Thermomagnetic Ettingshausen-Nernst effect in tachocline, magnetic reconnection phenomenon in lower layers, axion mechanism of solar luminosity variations, coronal heating problem solution and mechanism of ADM variations around BH

TL;DR

This paper proposes a holographic quantum gravity model explaining solar magnetic phenomena, including the Ettingshausen-Nernst effect, magnetic reconnection, and axion-related luminosity variations, offering solutions to longstanding solar physics problems.

Contribution

It introduces a holographic principle-based mechanism generating strong toroidal magnetic fields via the thermomagnetic EN effect, linking quantum gravity to solar magnetic and luminosity variations.

Findings

Holographic model predicts magnetic field neutralization in the Sun's core.

Thermomagnetic EN effect explains magnetic field generation in the tachocline.

Modulation of dark matter and axions influences solar activity cycles.

Abstract

It is shown that the holographic principle of quantum gravity (in the hologram of the Universe, and therefore in our Galaxy, and of course on the Sun!), in which the conflict between the theory of gravitation and quantum mechanics disappears, gives rise to the Babcock-Leighton holographic mechanism. Unlike the solar dynamo models, it generates a strong toroidal magnetic field by means of the thermomagnetic Ettingshausen-Nernst (EN) effect in the tachocline. Hence, it can be shown that with the help of the thermomagnetic EN effect, a simple estimate of the magnetic pressure of an ideal gas in the tachocline of e.g. the Sun can indirectly prove that by using the holographic principle of quantum gravity, the repulsive toroidal magnetic field of the tachocline ($B_{tacho}^{Sun} = 4.1 \cdot 10^7 ~G = - B_{core}^{Sun}$) precisely "neutralizes" the magnetic field in the Sun core, since the projections of the magnetic fields in the tachocline and the core have equal values but opposite directions. The basic problem is a generalized problem of the antidynamo model of magnetic flux tubes (MFTs), where the nature of both holographic effects (the thermomagnetic EN~effect and Babcock-Leighton holographic mechanism), including magnetic cycles, manifests itself in the modulation of asymmetric dark matter (ADM) and, consequently, the solar axion in the Sun interior.

Figures

c in Parker ([1955*b*](#bib.bib617))). In other words, how does this very cool region prevent the neutral atoms to penetrate from the upper convection zone to the base of the convection zone, i.e. the tachocline?

a in Chen and Bastian ([2012](#bib.bib173)); Blumenthal and Gould ([1970](#bib.bib124)); Tucker ([1975](#bib.bib798)); Brunetti ([2000](#bib.bib150)); Aharonian and Atoyan ([1981](#bib.bib13)); Krucker et al. ([2008](#bib.bib483))), where the energy distributions of electrons with different spectral indices are $\delta=1$ (based on the kinetic energy of the electron, which has a power-law form $\sim(\gamma-1)^{-\delta}$ (Chen and Bastian, [2012](#bib.bib173)). The energy of the incident EUV/SXR photon of axion origin is 4.2 keV (Fig. [22](#S3.F22)a; see analogous Fig. 5 in Chen and Bastian ([2

b in Fan ([2009](#bib.bib264))), which generally allows a balance of TEQ between four main forces: nonzero buoyancy, magnetic tension, aerodynamic drag, and Coriolis force.

a,b). Therefore, the values of the “experimental” data (on the left: Eq. ([110](#S3.E110))) are rather rough, while the theoretical data are more accurate.

b in Zioutas et al. ([2009](#bib.bib881))) become red near the edges, thus leading to the visible widening of the high intensity bands.

Equations

$$P_{a\rightarrow\gamma}=\left(\Delta_{a\gamma}L\right)^{2}\sin^{2}\left(\frac{\Delta_{osc}L}{2}\right)\Big{/}\left(\frac{\Delta_{osc}L}{2}\right)^{2}\,,$$

$${\Delta }_{osc}^2={\left({\Delta }_{pl}+{\Delta }_{Q,\perp }-{\Delta }_a\right)}^2+4{\Delta }_{a\gamma }^2$$

$${\Delta }_{a\gamma }=\frac{g_{a\gamma }B}{2}=540\left(\frac{g_{a\gamma }}{10^{-10}Ge{V}^{-1}}\right)\left(\frac{B}{1G}\right)p{c}^{-1},$$

$${\Delta }_a=\frac{m_a^2}{2E_a}=7.8\cdot 10^{-11}{\left(\frac{m_a}{10^{-7}eV}\right)}^2\left(\frac{10^{19}eV}{E_a}\right)p{c}^{-1},$$

$${\Delta }_{pl}=\frac{{\omega }_{pl}^2}{2E_a}=1.1\cdot 10^{-6}\left(\frac{n_e}{10^{11}c{m}^{-3}}\right)\left(\frac{10^{19}eV}{E_a}\right)p{c}^{-1},$$

$${\Delta }_{Q,\perp }=\frac{m_{\gamma ,\perp }^2}{2E_a}.$$

$$P_{a\to \gamma }\cong {\sin }^2\left({\Delta }_{a\gamma }L\right),$$

$$P_{a\to \gamma }\cong {\left(\frac{g_{a\gamma }BL}{2}\right)}^2.$$

$$L_{osc}=\frac{\pi }{{\Delta }_{a\gamma }}=\frac{2\pi }{g_{a\gamma }B}\cong 5.8\cdot 10^{-3}\left(\frac{10^{-10}Ge{V}^{-1}}{g_{a\gamma }}\right)\left(\frac{1G}{B}\right)pc$$

$$p_{ext}=p_{int}+\frac{|\vec{B}{|}^2}{8\pi },$$

$${\vec{j}}_{\perp }=\frac{3k{n}_{e}c}{2B^2}\vec{B}\times \nabla T$$

$${\vec{j}}_{\perp }=\frac{3knc}{2B^2}\frac{Z}{Z+1}\vec{B}\times \nabla T.$$

$$\vec{F}=\frac{1}{c}{\vec{j}}_{\perp }\times \vec{B}=\frac{3nk}{2B^2}\frac{Z}{Z+1}\left(\vec{B}\times \nabla T\right)\times \vec{B},$$

$$\vec{F}=\frac{3nk}{2}\frac{Z}{Z+1}\nabla T,$$

$$\vec{F}=\frac{1}{c}{\vec{j}}_{\perp }\times \vec{B}=\nabla p$$

$$\frac{3nk}{2}\frac{Z}{Z+1}\nabla T=nk\nabla T+kT\nabla n$$

$$a\frac{\nabla T}{T}+\frac{\nabla n}{n}=0,wherea=\frac{2-Z}{2(Z+1)},$$

$$T^{a}n=const.$$

$$T^{1/4}n=const.$$

$$j_{\perp }=j_y=-\frac{3nkc}{4B}\frac{dT}{dz}.$$

$$j_y=\frac{c}{4\pi }\frac{dB}{dz},$$

$$2B\frac{dB}{dz}=-6\pi kn\frac{dT}{dz}.$$

$$n=\frac{n_{tacho}{T}_{tacho}^{1/4}}{T^{1/4}},$$

$$d\left(B^2\right)=-\frac{6\pi k{n}_{tacho}{T}_{tacho}^{1/4}}{T^{1/4}}dT.$$

$$\frac{B_{tacho}^2}{8\pi }=n_{tacho}k{T}_{tacho},$$

$$\frac{B_{tacho}^2}{8\pi }=p_{ext}\approx 6.5\cdot 10^{13}\frac{erg}{c{m}^3}at0.7R_{Sun},$$

$$B_{tacho}\simeq 4100T=4.1\cdot 10^{7}G.$$

$$p_{ext}=\frac{B^2}{8\pi }$$

$$\frac{1}{k_R}=\underset{0}{\overset{\infty }{\int }}d\nu \frac{1}{k_{\nu }}\frac{d{B}_{\nu }}{dT}/\underset{0}{\overset{\infty }{\int }}d\nu \frac{d{B}_{\nu }}{dT},$$

$$p_{int}=nkT,where\{\begin{array}{l}T\simeq 10^{3.48}K,\\ T\simeq 10^{3.29}K,\\ T\simeq 10^{3.20}K,\\ T\simeq 10^{3.11}K,\end{array}\rho =10^{-7}g/c{m}^3.$$

Full text

The general laws of the theory of virtually empty anchored flux tubes with $B\sim 10^{7}~{}G$, which are born on the interface between the tachocline and the overshoot layer, are developed. These flux tubes are pushed out by means of the effective increase in their magnetic buoyancy – from the tachocline to the surface of the Sun. The appearance of MFTs on the surface of the Sun and the observed sunspots are identical. This means that the formation of the magnetic cycles of flux tubes coincides with the observational data of the Joy’s Law, and both effects are the manifestation of dark matter (DM) – solar axions in the core of the Sun, the modulation of which is predetermined by the anticorrelation modulation of ADM.

It is shown that the hypothesis of the axion mechanism of solar luminosity variations suggesting that the solar axion particles are born in the core of the Sun and may be efficiently converted back into $\gamma$-quanta in the magnetic field of the solar overshoot tachocline is physically relevant. As a result, it is also shown that the intensity variations of the $\gamma$-quanta of axion origin, induced by the magnetic field variations in the tachocline via the thermomagnetic EN effect, directly cause the Sun luminosity variations and eventually characterize the active and quiet states of the Sun. Within the framework of this mechanism estimations of the strength of the axion coupling to a photon ($g_{a\gamma}=4.4\cdot 10^{-11}~{}GeV^{-1}$) and the hadronic axion particle mass ($m_{a}\sim 3.2\cdot 10^{-2}~{}eV$) have been obtained. It is also shown that the claimed axion parameters do not contradict any known experimental and theoretical model-independent limitations.

The theoretical solution of the problem of coronal heating is suggested involving the photons of axion origin, which are produced in the core of the Sun. Our physical understanding of coronal heating and the coincidence of the theoretical and experimental photon spectra of the corona are connected with the appearance of a magnetic flux and simultaneously the flux of axion-origin photons in the outer layers of the Sun. On the other hand, it is connected with the basic mechanism of formation of sunspots and active regions, being an integral part of the solar cycle, which determines the corresponding variation in the release of energy in the corona and flares.

It is assumed that the modulation of the ADM density as a “clock” that anticorrelatively regulates the rate of the solar cycle, the photons of axion origin, the solar abundance and sunspots, and the mechanism of ADM density variations around the black hole (BH), is a consequence of the DM modulation in the isolated group of galaxies (with the active galactic nuclei (AGNs)), one of which is our Galaxy – the Milky Way. The main result of our model is the following one: the modulation of DM halo density at the Galactic Center (GC), which is closely correlated with the modulation of baryon matter density near supermassive black hole (SMBH), determines the appearance of the DM modulation by S-stars found at the GC. Based on empirical evidence, we make the following natural conclusion. If the modulations of the DM halo at the GC give rise to the modulations of the DM halo density at the surface of the Sun (through vertical density waves from the disk to the solar neighborhood), then there is an “experimental” anticorrelational relation between the DM density modulation in solar interior and the sunspot number. This is true for the relation between the periods of S-stars and sunspot number cycles.

1*Department of Theoretical and Experimental Nuclear Physics,

Odessa National Polytechnic University, Odessa, Ukraine*

2*CREST and NASA Research Centers, North Carolina Central University,

Durham, North Carolina, U.S.A.*

3*Department of Computer Technology and Applied Mathematics,

Kuban State University, Krasnodar, Russia*

1 Introduction

A hypothetical pseudoscalar particle called axion is predicted by the theory related to solving the CP-invariance violation problem in QCD. The most important parameter determining the axion properties is the energy scale $f_{a}$ of the so-called U(1) Peccei-Quinn symmetry violation. It determines both the axion mass and the strength of its coupling to fermions and gauge bosons including photons. However, in spite of the numerous direct experiments, axions have not been discovered so far. Meanwhile, these experiments together with the astrophysical and cosmological limitations leave a rather narrow band for the permissible parameters of invisible axion (e.g. $10^{-6}eV\leqslant m_{a}\leqslant 10^{-2}eV$ (Raffelt, 2004, 2008)), which is also a well-motivated cold dark matter (CDM) candidate in this mass region (Raffelt, 2004, 2008).

A whole family of axion-like particles (ALPs) may exist along with axions, having the similar Lagrangian structure relative to the Peccei-Quinn axion, as well as their own distinctive features. If they exist, the connection between their mass and their constant of coupling to photons may be highly weakened, as opposed to the axions. It should be also mentioned that the phenomenon of photon-ALP mixing in the presence of the electromagnetic field not only leads to the classic neutrino-like photon-ALP oscillations, but also causes the change in the polarization state of the photons propagating in the strong enough magnetic fields (the $a\gamma\gamma$ coupling acts like a polarimeter (De Angelis et al., 2011)). It is generally assumed that there are light ALPs coupled only to photons, although the realistic models of ALPs with couplings both to photons and to matter are not excluded (Masso, 2008). Anyway, they may be considered a well-motivated CDM candidate (Raffelt, 2004, 2008) under certain conditions, just like axions.

It is interesting to note that the photon-ALP mixing in magnetic fields of different astrophysical objects including active galaxies, clusters of galaxies, intergalactic space and the Milky Way, may be the cause of the remarkable phenomena like dimming of stars luminosity (e.g. supernovae in the extragalactic magnetic field (Csaki et al., 2002; Mirizzi et al., 2005)) and “light shining through a wall” (e.g. light from very distant objects, traveling through the Universe (De Angelis et al., 2011; Fairbairn et al., 2011)). In the former case the luminosity of an astrophysical object is dimmed because some part of photons transforms into axions in the object’s magnetic field. In the latter case photons produced by the object are initially converted into axions in the object’s magnetic field, and then after passing some distance (the width of the “wall”) are converted back into photons in another magnetic field (e.g. in the Milky Way), thus emulating the process of effective free path growth for the photons in astrophysical medium (Adler et al., 2008; Redondo and Ringwald, 2010).

For the sake of simplicity let us hereinafter refer to all such particles as axions if not stated otherwise.

In the present paper we consider the possible existence of the axion mechanism of Sun luminosity variations222Let us point out that the axion mechanism of Sun luminosity used for estimating the axion mass was described for the first time in 1978 by Vysotsskii et al. (1978). based on the “light shining through a wall” effect. To be more precise, we attempt to explain the axion mechanism of Sun luminosity variations by the “light shining through a wall”, when the photons born mainly in the solar core are at first converted into axions via the Primakoff effect (Primakoff, 1951) in its magnetic field, and then are converted back into photons after passing the solar radiative zone and getting into the magnetic field of the overshoot tachocline (see e.g. Fig. 1 in Zioutas et al. (2009)).

It gives rise to the very intriguing questions: “What is the nature of the tachoclines on the Sun and other stars? If the existence of the tachocline is predetermined by the holographic principle of quantum gravity, then why, in contrast to the solar dynamo, the magnetic field of the tachocline is much stronger than $10^{5}~{}G$? What is the strength of the magnetic field in the Sun core? What is it in the Sun core that controls the solar cycle? Can axions and/or DM particles in the Sun core control the solar cycle of the photons of axion origin, the abundance of Sun interior and sunspots?”

Based on the solution of such nontrivial physics, we estimate this magnetic field within the framework of the thermomagnetic EN effect (as a consequence of the fundamental properties of the holographic principle of quantum gravity). In addition, we obtain the consistent estimates for the axion mass ($m_{a}$) and the axion coupling constant to photons ($g_{a\gamma}$), based on this mechanism, and verify their values against the axion model results and the known experiments including CAST, ADMX, RBF.

2 Photon-axion conversion and the case of maximal mixing

Let us give some implications and extractions from the photon-axion oscillations theory which describes the process of the photon conversion into an axion and back under the constant magnetic field $B$ of the length $L$. It is easy to show (Fairbairn et al., 2011; Raffelt and Stodolsky, 1988; Mirizzi et al., 2005; Hochmuth and Sigl, 2007) that in the case of the negligible photon absorption coefficient ($\Gamma_{\gamma}\to 0$) and axions decay rate ($\Gamma_{a}\to 0$) the conversion probability is

[TABLE]

where the oscillation wavenumber $\Delta_{osc}$ is given by

[TABLE]

while the mixing parameter $\Delta_{a\gamma}$, the axion-mass parameter $\Delta_{a}$, the refraction parameter $\Delta_{pl}$ and the QED dispersion parameter $\Delta_{Q,\perp}$ may be represented by the following expressions:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here $g_{a\gamma}$ is the constant of axion coupling to photons; $B$ is the transverse magnetic field; $m_{a}$ and $E_{a}$ are the axion mass and energy; $\omega^{2}_{pl}=4\pi\alpha n_{e}/m_{e}$ is an effective photon mass in terms of the plasma frequency if the process does not take place in vacuum, $n_{e}$ is the electron density, $\alpha$ is the fine-structure constant, $m_{e}$ is the electron mass; $m_{\gamma,\perp}^{2}$ is the effective mass square of the transverse photon which arises due to interaction with the external magnetic field.

The conversion probability (1) is energy-independent, when $2\Delta_{a\gamma}\approx\Delta_{osc}$, i.e.

[TABLE]

or whenever the oscillatory term in (1) is small ($\Delta_{osc}L/2\to 0$), implying the limiting coherent behavior

[TABLE]

It is worth noting that the oscillation length corresponding to (7) reads

[TABLE]

assuming a purely transverse field. In the case of the appropriate size $L$ of the region a complete transition between photons and axions is possible.

From now on we are going to be interested in the energy-independent case (7) or (8) which plays the key role in determination of the parameters for the axion mechanism of Sun luminosity variations hypothesis (the axion coupling constant to photons $g_{a\gamma}$, the transverse magnetic field $B$ of length $L$ and the axion mass $m_{a}$).

3 Axion mechanism of Sun luminosity variations

Our hypothesis is that the solar axions, which are born in the solar core (Raffelt, 2004, 2008) through the known Primakoff effect (Primakoff, 1951), may be converted back into $\gamma$-quanta in the magnetic field of the solar tachocline (the base of the solar convective zone). The magnetic field variations in the tachocline cause the converted $\gamma$-quanta intensity variations in this case, which in their turn cause the variations of the Sun luminosity known as the active and quiet Sun states. Let us consider this phenomenon in more detail.

As we noted above, the expression (1) for the probability of the axion-photon oscillations in the transversal magnetic field was obtained for the media with quasi-zero refraction, i.e. for the media with a negligible photon absorption coefficient ($\Gamma_{\gamma}\to 0$). It means that in order for the axion-photon oscillations to take place without any significant losses, a medium with a very low or quasi-zero density is required, which would suppress the processes of photon absorption almost entirely.

Surprisingly enough, it turns out that such “transparent” media can take place, and not only in plasmas in general, but straight in the convective zone of the Sun. Here we generally mean the so-called MFTs, the properties of which are examined below.

3.1 Ideal photon channeling conditions inside the magnetic flux tubes

The idea of the energy flow channeling along a fanning magnetic field has been suggested for the first time by Hoyle (1949) as an explanation for darkness of umbra of sunspots. It was incorporated in a simple sunspot model by Chitre (1963). Zwaan (1978) extended this suggestion to smaller flux tubes to explain the dark pores and the bright faculae as well. Summarizing the research of the convective zone magnetic fields in the form of isolated flux tubes, Spruit and Roberts (1983) suggested a simple mathematical model for the behavior of thin MFTs, dealing with the nature of the solar cycle, sunspot structure, the origin of spicules and the source of mechanical heating in the solar atmosphere. In this model, the so-called thin tube approximation is used (see Spruit and Roberts (1983) and Refs. therein), i.e. the field is conceived to exist in the form of slender bundles of field lines (flux tubes) embedded in a field-free fluid (Fig. 1). Mechanical equilibrium between the tube and its surrounding is ensured by a reduction of the gas pressure inside the tube, which compensates the force exerted by the magnetic field. In our opinion, this is exactly the kind of mechanism Parker (1955b) was thinking about when he wrote about the problem of flux emergence: “Once the field has been amplified by the dynamo, it needs to be released into the convection zone by some mechanism, where it can be transported to the surface by magnetic buoyancy” (Hassan, 2003).

In order to understand magnetic buoyancy, let us consider an isolated horizontal flux tube in pressure equilibrium with its non-magnetic surroundings, so that in cgs units

[TABLE]

where $p_{int}$ and $p_{ext}$ are the internal and external gas pressures, respectively, while $B$ denotes the uniform field strength in the flux tube. If the internal and external temperatures are equal, so that $T_{int}=T_{ext}$ (thermal equilibrium), then since $p_{ext}>p_{int}$, the gas in the tube is less dense than its surrounding ($\rho_{ext}>\rho_{int}$), implying that the tube will rise under the influence of gravity.

In spite of the obvious, though turned out to be surmountable, difficulties of the application to real problems, it was shown (see Spruit and Roberts (1983) and Refs. therein) that strong buoyancy forces act in MFTs of the required field strength ($10^{4}-10^{5}~{}G$ (Ruzmaikin, 2000)). Under their influence tubes either float to the surface as a whole (e.g. Fig. 1 in Fisher et al. (2000)) or they form loops of which the tops break through the surface (e.g. Fig. 1 in Zwaan (1978)) and lower parts descend to the bottom of the convective zone, i.e. to the overshoot tachocline zone. The convective zone, being unstable, enhanced this process (Spruit and van Ballegooijen, 1982; Golub et al., 1981). Small tubes take longer to erupt through the surface because they feel stronger drag forces. It is interesting to note here that the phenomenon of the drag force, which raises the magnetic flux tubes to the convective surface with the speeds about 0.3-0.6 km/s, was discovered in direct experiments using the method of time-distance helioseismology (Ilonidis et al., 2011). Detailed calculations of the process (Moreno-Insertis, 1983) show that even a tube with the size of a very small spot, if located within the convective zone, will erupt in less than two years. Yet, according to Moreno-Insertis (1983), horizontal fields are needed in the overshoot tachocline zone, which survive for about 11 yr, in order to produce an activity cycle.

A simplified scenario of MFTs birth and space-time evolution (Fig. 1a) can be presented as follows. A MFT is born in the overshoot tachocline zone (Fig. 1c) and rises up to the convective zone surface (Fig. 1b) without separation from the tachocline (the anchoring effect), where it forms a sunspot (Fig. 1d) or other kinds of active solar regions when intersecting the photosphere. There are more fine details of MFT physics expounded in overviews by Hassan (2003) and Fisher et al. (2000), where certain fundamental questions, which need to be addressed to understand the basic nature of magnetic activity, are discussed in detail: How is the magnetic field generated, maintained and dispersed? What are its properties such as structure, strength, geometry? What are the dynamical processes associated with magnetic fields? What role do magnetic fields play in the energy transport?

Dwelling on the last extremely important question associated with the energy transport, let us note that it is known that thin MFTs can support longitudinal (also called sausage), transverse (also called kink), torsional (also called torsional Alfvén), and fluting modes (see e.g. Spruit (1982); Hollweg (1990); Roberts (1991); Roberts and Ulmschneider (1997); Stix (2004)); for the tube modes supported by wide MFTs, see Roberts and Ulmschneider (1997). Focusing on the longitudinal tube waves known to be an important heating agent of solar magnetic regions, it is necessary to mention the recent papers by Fawzy et al. (2012), which showed that the longitudinal flux tube waves are identified as insufficient to heat the solar transition region and corona, in agreement with the previous studies (Fawzy and Cuntz, 2011). In other words, the problem of generation and transport of energy by MFTs remains unsolved in spite of its key role in physics of various types of solar active regions.

It is clear that this unsolved problem of energy transport by magnetic flux tubes at the same time represents another unsolved problem related to the energy transport and sunspot darkness (see 2.2 in Rempel and Schlichenmaier (2011)). From a number of known concepts playing a noticeable role in understanding of the connection between the energy transport and sunspot darkness, let us consider the most significant theory, according to our vision. It is based on the Parker-Biermann cooling effect (Parker, 1955a; Biermann, 1941; Parker, 1979b) and originates from the works of Biermann (1941) and Alfvén (1942).

The main point of the Parker-Biermann cooling effect is that the classical mechanism of the magnetic tubes buoyancy (see e.g. Fig. 2a, Parker (1955a)), emerging as a result of the shear flows instability development in the tachocline, should be supplemented with the following results of the Biermann (1941) postulate and the theory developed by Parker (1955a, 1979b): the electric conductivity in the strongly ionized plasma may be so high that the magnetic field becomes frozen into plasma and causes the split magnetic tube (Fig. 2b,c) to cool inside.

Biermann understood that the magnetic field within the sunspots might itself be a reason of their darkness. Around the sunspots, the heat is transported up to the surface of the Sun by means of convection (see 2.2.1 in Rempel and Schlichenmaier (2011)), while Biermann (1941) noted that such transport is strongly inhibited by the nearly vertical magnetic field within the sunspot, thereby providing a direct explanation for the reduced temperature at the visible surface. Thus, the sunspot is dark because it is cooler than its surroundings, and it is cooler because the convection is inhibited underneath.

Still, the missing cause of a very high conductivity in strongly ionized plasma, which would produce a strong magnetic field “frozen” into this plasma, has been the major flaw of the so-called Biermann (1941) postulate.

Let us show a solution to the known problem of the Parker-Biermann cooling effect, which is defined by the nature of very large toroidal magnetic fields in the tachocline (determined by the thermomagnetic EN effect) and provides the physical basis for the photon channeling conditions inside the MFTs.

3.1.1 Thermomagnetic Ettingshausen-Nernst effect and the nature of toroidal magnetic field in the tachocline

It is known (Schwarzschild, 1958) that the temperature dependence of the thermonuclear reaction rate is proportional to T4.5 in the 107 K region. It means there is a sharp boundary between the hot region embracing the majority of thermonuclear reactions and a cooler one with virtually no thermonuclear reactions at all (Winterberg, 2015). This boundary between the radiation zone and the convection zone is the tachocline. Because of the great temperature gradient in the tachocline, the thermomagnetic EN effect (Ettingshausen and Nernst, 1886; Sondheimer, 1948; Spitzer, 1956; Kim and Stephen, 1969) produces strong electric currents screening the intense magnetic fields $\sim 5\cdot 10^{7}$ G of the solar core (Fowler et al., 1955; Couvidat et al., 2003).

Let us note that the thermomagnetic effect described by Eqs. (5-49) and (5-52) in Spitzer (1956, 1962, 2006) is often called the Nernst effect. This is not entirely a correct name, since the similar equations were experimentally obtained earlier in 1886 by A.V. Ettingshausen, the teacher, with his young student W. Nernst helping him (see Ettingshausen and Nernst (1886); Sondheimer (1948)). Therefore let us hereinafter refer to this effect as the thermomagnetic EN effect (see 3.1.1 in Rusov et al. (2015)).

The EN effect theory and experiments are known to be the basis for the quantum critical phenomena and kinetic fluctuations (see e.g. Hertz (1976); Coleman and Schofield (2005); Michaeli and Finkel’stein (2009)) in the 2D materials under magnetic fields. Let us recall e.g. the superconductors (Xu et al., 2000; Wang et al., 2006), the amorphous superconducting films (Michaeli and Finkel’stein, 2009), semiconductor heterostructures and graphene (Bergman and Oganesyan, 2010), and what is not surprising taking into account the “quantum conundrum” (Hertz, 1976; Zaanen, 2007) – the dyonic BHs (Hartnoll et al., 2007).

The bright idea of Winterberg (2015, 2016) to confine thermonuclear plasma by means of the repulsive gravitational field in an ultracentrifuge using the Einstein’s general theory of relativity and the repulsive magnetic field induced by the thermomagnetic effect prompts that it is the repulsive quantum gravitation in 2D layer which gives rise to the repulsive magnetic field induced by the thermomagnetic EN effect. This may happen e.g. in the tachocline of the Sun or white dwarfs (see Rusov et al. (2015)) or in dyonic BHs (Hartnoll et al., 2007)).

A natural question arises here: can quantum gravity in 2D materials (which is related to gravity in 3D anti-de Sitter (AdS) space (see Hartnoll et al. (2007))) remove the discrepancies between the general theory of relativity and quantum mechanics (more exactly, the quantum field theory on the boundary of the asymptotically AdS space) by means of the known “holographic renormalization” (see Skenderis (2002); Papadimitriou (2016)), and give rise to the thermomagnetic EN effect at the same time? Let us roughly outline this below.

\subsubsubsection

Thermomagnetic Ettingshausen-Nernst effect of toroidal magnetic field in the tachocline

The thermomagnetic current can be generated in the magnetized plasma under the quasi-steady magnetic field in the weak collision approximation (the collision frequency much less than the positive ion cyclotron frequency) (Spitzer, 1962, 2006). For the fully ionized plasma the EN effect yields the current density (see Eqs. (5-49) and (5-52) in Spitzer (1962, 2006)):

[TABLE]

where $n_{e}$ is the electron number density, $B$ is the magnetic field, $T$ is the absolute temperature, $k$ and $c$ stand for the Boltzmann constant and the speed of light, respectively. When $n_{e}=\left[Z/(Z+1)\right]n$, where $n=n_{e}+n_{i}$, and $n_{i}=n_{e}/Z$ is the ion number density for a $Z$-times ionized plasma,

[TABLE]

It exerts a force on plasma, with the force density $F$ given by

[TABLE]

or with $\nabla T$ perpendicular to $\vec{B}$

[TABLE]

which leads to the magnetic equilibrium (see Eq. (4-1) in Spitzer (1962)):

[TABLE]

with $p=nkT$. By equating (14) and (15),

[TABLE]

or

[TABLE]

we obtain the condition

[TABLE]

For the singly ionized plasma with $Z=1$,

[TABLE]

For the doubly ionized plasma ($Z=2$) $n=const$. Finally, in the limit of large $Z$, $T^{-1/2}n=const$, and $n$ does not depend on $T$ strongly, as opposed to the case of the plasma at a constant pressure with $Tn=const$. Thus, the thermomagnetic currents can change the pressure distribution in the magnetized plasma considerably.

Choosing the Cartesian coordinate system with $z$ axis along $\nabla T$, $x$ axis along the magnetic field and $y$ axis along the current, and assuming the fully ionized hydrogen plasma with $Z=1$ in the tachocline, we obtain

[TABLE]

From Maxwell’s equation $4\pi\vec{j}_{\perp}/c=\nabla\times\vec{B}$, one has

[TABLE]

then by equating (20) and (21) we derive

[TABLE]

From (19) one has

[TABLE]

where the values $n=n_{tacho}$ and $T=T_{tacho}$ correspond to the overshoot tachocline. Substituting (23) into (22), we find

[TABLE]

As a result of integrating in the limits $[B_{tacho},0]$ on the left and $[0,T_{tacho}]$ on the right,

[TABLE]

expressing the fact that the magnetic field of the thermomagnetic current in the overshoot tachocline “neutralizes” the magnetic field of the solar interior completely, so that the projections of the magnetic field in the tachocline and in the core are equal but of opposite directions (see Fig. 3).

An intriguing question arises here of what forces are the cause of the shielding of the strong magnetic fields of the solar core and the radiation zone, which would be related to the enormous magnetic pressure in the overshoot tachocline (see Fig. 3),

[TABLE]

where the gas pressure $p_{ext}$ in the solar tachocline ($\rho\approx 0.2~{}g\cdot cm^{-3}$ and $T\approx 2.3\cdot 10^{6}K$ (Bahcall and Pinsonneault, 1992) at $0.7R_{Sun}$) yields the toroidal magnetic field

[TABLE]

Propagating the question above even further, one may ask: “What kind of mysterious nature of the solar tachocline gives birth to the repulsive magnetic field through the thermomagnetic EN effect?” And finally, “If the thermomagnetic EN effect exists in the tachocline, what is its physical nature?” The essence of this physics will be discussed below.

\subsubsubsection

Nature of toroidal magnetic field of the tachocline in the interior of the Sun

The problem is devoted to the study of physics and the magnitude of the toroidal magnetic field in the tachocline. It is known that the radiation zone of the Sun rotates approximately as a solid body, and the convection zone has a differential rotation. This leads to the formation of a very strong shear layer between these two zones, which is called the tachocline. This gives rise to a fairly simple and at the same time very complicated question: “What is the nature and the consequence of the existence of tachoclines on the Sun or other stars?”

We propose a very unexpected answer that the existence of a two-dimensional surface of the tachocline in solar interior is the manifestation of the holographic principle in the Universe and, therefore, on the Sun. It is important to note that the holographic theory correlates the physical laws that act in some volume with the laws that act on the surface that limits this volume. The physics at the boundary is represented by quantum particles that have “colored” charges and interact almost like quarks and gluons in standard particle physics. The laws in the volume are a kind of string theory that includes the force of gravity (see Maldacena (2005)), which is difficult to describe in terms of quantum mechanics. The main result of the holographic principle is the fact that surface physics and physics in the volume are completely equivalent, in spite of the completely different ways of describing them.

In most situations, the contradictory requirements of quantum mechanics and general relativity are not a problem, because either quantum or gravitational effects are so small that they can be neglected. However, with a strong curvature of space-time, the quantum aspects of gravity become essential, and the conflict between the theory of gravity and quantum mechanics should disappear (see ’t Hooft (1993); Susskind (1995); Maldacena (1999); Hanada et al. (2014)). To create a large curvature of space-time, a very large mass or density is required. Some physicists believe that even the Sun is incapable of distorting space-time so that the manifestations of quantum gravity effects become obvious.

Unlike other physicists, we believe that the fundamental holographic principle of quantum gravity predicts the experimental possibility of observational measurements of magnetic fields between the two-dimensional surface of the tachocline and the three-dimensional volume of the core on compact objects – our Sun, magnetic white dwarfs, accreting neutron stars and BHs.

What does the equivalence of the two theories mean? First of all, for each object in one theory there must be an analog in the other (i.e. in quantum mechanics and gravitation). Descriptions of objects can be completely different: a certain particle within a volume can correspond to a whole set of particles on its boundary, considered as a single entity. Second, the predictions for the corresponding objects must be identical. For example, if two particles in the volume collide with a probability of 40%, then the corresponding aggregate of particles on its boundary should also be confronted with a probability of 40% (see Maldacena (2005)).

Hence, it is not difficult to show that with the help of the thermomagnetic EN effect, a simple estimate of the magnetic pressure of an ideal gas in the tachocline of e.g. the Sun can indirectly prove that by using the holographic principle of quantum gravity, the repelling toroidal magnetic field of the tachocline ($B_{tacho}^{Sun}=4.1\cdot 10^{7}~{}G=-B_{core}^{Sun}$; see Eq. (27) and Rusov et al. (2015)) exactly “neutralizes” the magnetic field in the Sun’s core, where the projections of the magnetic fields of the tachocline and the core have the equal value but the opposite directions.

This means that on the one hand, the holographic Babcock-Leighton (BL) mechanism (see Fig. 4), which we often refer to as the holographic antidynamo mechanism, is caused by a remarkable example of the Cowling antidynamo theorem. This theorem states that no axisymmetric magnetic field can be maintained through the self-sustaining action of the dynamo by means of an axially symmetric current (Cowling, 1933). On the other hand, the BL mechanism follows our example of a thermomagnetic EN effect, or the so-called solar holographic antidynamo, in which the poloidal field originates directly from the toroidal field, as shown in Fig. 4b, but not vice versa (see Fig. 4a).

But from here we also understand that the remarkable properties of the holographic BL mechanism are the consequence of not only the fundamental features of the holographic principle of quantum gravity, but also of DM. The existence and the true nature of DM are predetermined, surprisingly enough, by the law of quantum-gravitational energy conservation in the Universe, and therefore, in our Galaxy, and of course, in the Sun, in which DM is actively captured by the solar gravitational field!

The main examples of the DM existence in the Sun and other stars in our Galaxy will be considered in our work.

3.1.2 Parker-Biermann cooling effect, Rosseland mean opacity and

axion-photon oscillations in twisted magnetic tubes

Several local models are known to have been used with great success to investigate the formation of buoyant magnetic transport, which transforms twisted magnetic tubes generated through shear amplification near the base tachocline (see e.g. Nelson and Miesch (2014)) and the structure and evolution of photospheric active regions (see e.g. Rempel (2011)).

Because these models assume the anchored MFTs depending on the poloidal field in the tachocline, it is not too hard to show that the magnetic field $B_{tacho}$ reaching $\sim 4100$ T (see (27)) may be at the same time the reason for the Parker-Biermann cooling effect in the twisted magnetic tubes (see Fig. 5b). The theoretical consequences of such reasoning of the Parker-Biermann cooling effect are considered below.

First of all, we suggest that the classic mechanism of magnetic tubes buoyancy (Fig. 5a), appearing as a result of the shear instability development in the tachocline, should be supplemented by the rise of the twisted magnetic tubes in a stratified medium (Fig. 5b; see Fig. 1 and Fig. 2 in Sturrock et al. (2001)), where the magnetic field is produced by antidynamo action throughout the convection zone, primarily by stretching and twisting in the turbulent downflows (see Stein (2012b)).

Second, the twisting of the magnetic tube can not only promote its splitting, but also form a cool region under a certain condition

[TABLE]

when the tube (inset in Fig. 5b) is in thermal disequilibrium with its surroundings and the convective heat transfer is suppressed (Biermann, 1941).

It is interesting to explore how the cool region stretching from the tachocline to the photosphere, where the magnetic tube is in thermal non-equilibrium (28) with its surroundings, deals with the appearance of neutral atoms (e.g. hydrogen) in the upper convection zone (see Fig. 6a in contrast to Fig. 2c in Parker (1955b)). In other words, how does this very cool region prevent the neutral atoms to penetrate from the upper convection zone to the base of the convection zone, i.e. the tachocline?

It is essential to find the physical solution to the problem of solar convective zone which would fit the opacity experiments. The full calculation of solar opacities, which depend on the chemical composition, pressure and temperature of the gas, as well as the wavelength of the incident light, is a complex endeavor. The problem can be simplified by using the mean opacity averaged over all wavelengths, so that only the dependence on the gas physical properties remains (see e.g. Rogers and Iglesias (1994); Ferguson et al. (2005); Bailey et al. (2009)). The most commonly used is the Rosseland mean opacity $k_{R}$, defined as:

[TABLE]

where $dB_{\nu}/dT$ is the derivative of the Planck function with respect to temperature, $k_{\nu}$ is the monochromatic opacity at frequency $\nu$ of the incident light or the total extinction coefficient, including stimulated emission plus scattering. A large value of the opacity indicates strong absorption from beam of photons, whereas a small value indicates that the beam loses very little energy as it passes through the medium.

Note that the Rosseland opacity is the harmonic mean, in which the greatest contribution comes from the lowest values of opacity, weighted by a function that depends on the rate at which the blackbody spectrum varies with temperature (see Eq. (29) and Fig. 7), and the photons are most efficiently transported through the “windows” where $k_{\nu}$ is the lowest (see Fig. 2 in Bailey et al. (2009)).

Taking the Rosseland mean opacities shown in Fig. 7, one may calculate, for example, four consecutive cool ranges within the convective zone (Fig. 6a), where the internal gas pressure $p_{int}$ is defined by the following values:

[TABLE]

Since the inner gas pressure (30) grows towards the tachocline so that

[TABLE]

it becomes evident that the neutral atoms appearing in the upper convection zone ($\geqslant 0.85R_{Sun}$) cannot descend deep to the base of the convection zone, i.e. the tachocline (see Fig. 6a).

Therefore, it is very important to examine the connection between the Rosseland mean opacity and axion-photon oscillations in a twisted magnetic tube.

Let us consider the qualitative nature of the $\Omega$-loop formation and growth process, based on the semiphenomenological model of the magnetic $\Omega$-loops in the convective zone.

$\bullet$ A high concentration azimuthal magnetic flux ($B_{tacho}\sim 4100$ T, see Fig. 6) in the overshoot tachocline develops through the shear flows instability.

An interpretation of such a link is related to the fact that helioseismology places the principal rotation $\partial\omega/\partial r$ of the Sun in the overshoot layer immediately below the bottom of the convective zone (Parker, 1994). It is also generally believed that the azimuthal magnetic field of the Sun is produced by the shearing $r\partial\omega/\partial r$ of the toroidal field $B_{tacho}$, from which it is generally concluded that the principal azimuthal magnetic flux resides in the shear layer (Parker, 1955b, 1993).

$\bullet$ If some “external” factor of the local shear perturbation appears against the background of the azimuthal magnetic flux concentration, such an additional local density of magnetic flux can lead to the magnetic field strength as high as e.g. $B_{z}\sim 3600$ T (see Fig. 6a and Fig. 8b). Of course, this brings up a question about the physics behind such an “external” factor and the local shear perturbation.

In this regard let us consider the superintense magnetic $\Omega$-loop formation in the overshoot tachocline through the local shear caused by the high local concentration of azimuthal magnetic flux. The buoyant force acting on the $\Omega$-loop decreases slowly with concentration, so the vertical magnetic field of the $\Omega$-loop reaches $B_{z}\sim 3600$ T at about $R/R_{Sun}\sim 0.72$ (see Fig. 6a and Fig. 8b). Because of the magnetic pressure (see analogous Eq. (26) and Fig. 6a) $p_{ext}=B_{0.72R_{Sun}}^{2}/8\pi=5.21\cdot 10^{13}~{}erg/cm^{3}$ (Bahcall and Pinsonneault, 1992), this leads to significant cooling of the $\Omega$-loop tube (see Fig. 6a).

In other words, we assume the effect of the $\Omega$-loop cooling to be the basic effect responsible for the magnetic flux concentration. It arises from the well known suppression of convective heat transport by a strong magnetic field (Biermann, 1941). It means that although the principal azimuthal magnetic flux resides in the shear layer, it predetermines the additional local shear giving rise to significant cooling inside the $\Omega$-loop.

Thus, the ultralow pressure is set inside the magnetic tube as a result of the sharp limitation of the magnetic steps buoyancy inside the cool magnetic tube (Fig. 6a). This happens because the buoyancy of the magnetic flows requires finite superadiabaticity of the convection zone (Fan and Fisher, 1996; Fan, 2009), otherwise, expanding according to the magnetic adiabatic law (with the convection being suppressed by the magnetic field), the magnetic clusters may become cooler than their surroundings, which compensates the effect of the magnetic buoyancy of superintense magnetic O-loop.

Eventually we suppose that the axion mechanism based on the X-ray channeling along the “cool” region of the split magnetic tube (Fig. 6a) effectively supplies the necessary energy flux “channeling” in the magnetic tube to the photosphere while the convective heat transfer is heavily suppressed.

In this context it is necessary to have a clear view of the energy transport by X-rays of axion origin, which are a primary transfer mechanism. The recent improvements in the calculation of the radiative properties of solar matter have helped to resolve several long-standing discrepancies between the observations and the predictions of theoretical models (see e.g. Rogers and Iglesias (1994); Ferguson et al. (2005); Bailey et al. (2009)), and now it is possible to calculate the photon mean free path (Rosseland length) for Fig. 7:

[TABLE]

where the Rosseland mean opacity values $k_{R}$ and density $\rho$ are chosen so that the very low internal gas pressure $p_{int}$ (see Eq. (31)) along the entire magnetic tube almost does not affect the external gas pressure $p_{ext}$ (see (32) and Fig. 7).

Let us now examine the appearance of the X-rays of axion origin, induced by the magnetic field variations near the tachocline (Fig. 6a), and their impact on the Rosseland length (see (32)) inside the cool region of the magnetic tubes.

Let us remind that the magnetic field strength $B_{tacho}\sim 4100$ T in the overshoot tachocline (see Fig. 6a) and the Parker-Biermann cooling effect (see (28)) lead to the corresponding value of the magnetic field strength $B(0.72R_{Sun})\sim 3600$ T (see Fig. 6a), which in its turn implies the virtually zero internal gas pressure of the magnetic tube.

As it is shown above (see Priest and Forbes (2000)), the topological effect of the magnetic reconnection inside the $\Omega$-loop results in the formation of the so-called O-loops (Fig. 5 and Fig. 6a) with their buoyancy limited from above by the strong cooling inside the $\Omega$-loop (Fig. 6a). It is possible to derive the value of the horizontal magnetic field of the magnetic steps at the top of the O-loop: $B_{MS}\approx B(0.72R_{Sun})\sim 3600$ T.

So in the case of the large enough Rosseland length (see Eq. (32)), X-rays of axion origin, induced by the horizontal magnetic field in O-loops, reach the photosphere freely, while in the photosphere itself, according to the Rosseland length

[TABLE]

these photons undergo multiple Compton scattering producing a typical directional pattern (Fig. 6a).

Aside from the X-rays of axion origin with mean energy of $4.2~{}keV$ (Andriamonje et al., 2007; Zioutas et al., 2009), there are also $h\nu\sim 0.95~{}keV$ X-rays (originating from the interface between the radiation zone and overshoot tachocline, according to a theoretical estimate by Bailey et al. (2009)). Such X-rays would produce the Compton-scattered photons with mean energy of $\leqslant 0.95~{}keV$. These photons “disappear” by inverse X-ray transformation into axions (see Fig. 6, Fig. 9a,b and Fig. 11a). This way the $\sim 0.95~{}keV$ X-rays do not contradict the known measurements of the photons with mean energy of $\sim 4~{}keV$ (see Fig. 1 in Andriamonje et al. (2007) and Rieutord (2014)) by involving the X-rays of axion origin in O-loops (see Figs. 6 and 9a,b).

And finally, let us emphasize that we have just shown a theoretical possibility of the time variation of the sunspot activity to correlate with the flux of the X-rays of axion origin; the latter being controlled by the magnetic field variations near the overshoot tachocline. As a result, it may be concluded that the axion mechanism for solar luminosity variations based on the lossless X-ray “channeling” along the magnetic tubes allows to explain the effect of the almost complete suppression of the convective heat transfer, and thus to understand the known puzzling darkness of the sunspots (Rempel and Schlichenmaier, 2011).

3.1.3 Effect of virtually empty magnetic flux tubes, magnetic reconnection phenomenon and emergence of reconnecting flux-rope dynamo

The appearance of sunspots on the solar surface is one of the major manifestations of solar activity demonstrating the cyclic behavior with the period of about 11 years (see e.g. Hathaway (2015)). A very high density of the magnetic field is observed within the sunspots, which suppresses the convective heat flow from the solar interior to the surface (Biermann, 1941; Cowling, 1953; Parker, 1955a). That is why the sunspots are cooler and darker against the background of the solar disk. More than a hundred years ago Hale (1908) discovered the vertical “vortices” of the magnetic field in the sunspots. About a year later British astronomer Evershed (1909) was conducting the observations in Kodaikanal (Tamil Nadu, India) and found that in spite of the Hale’s vertical “vortices”, the magnetic field in the penumbra is radially divergent from the center of a sunspot. The mechanism of the sunspot formation including the umbra and penumbra as well as the Evershed effect are still the subject of active discussions and studies today, and lots of fundamental questions remain unanswered (Solanki, 2003; Borrero and Ichimoto, 2011; Tiwari et al., 2015; Pozuelo et al., 2015). We consider some possible solutions to these problems below.

We are mostly interested in effects for which the current theories, assuming that the sunspots are produced by the dynamo action at the bottom of the convection zone, fail to provide the convincing proofs and explain the dynamo action in the convection zone. This problem becomes even stronger with the recent findings. The numerical simulations of the solar dynamo have not revealed thin MFTs of the comparable strength so far (see e.g. Guerrero and Käpylä (2011); Nelson et al. (2013); Käpylä et al. (2013)). The helioseismology does not give any evidence of the upward MFTs existence either (Birch et al., 2013, 2016). This is especially important because of the recent helioseismological investigations which set the strict limitations on the velocities of large-scale convection inside the Sun and demonstrated the inconsistency with the existing global magnetoconvection (e.g. Stein, 2012a) modeling (Hanasoge et al., 2010, 2012, 2015; Gizon and Birch, 2012; Birch et al., 2013, 2016).

Thus, our major question is: “How are the sunspots generated by the strong magnetic field at the base of the convection zone without any dynamo action?”, or otherwise “Which fundamental processes connect the sunspot cycle with the large-scale magnetic field of the Sun?”

A possible answer to this question is that the properties of the tachocline, bringing the most important thermomagnetic EN effect, play the key role in the process of the strong magnetic field variation (without any large-scale dynamo!), and this variation is at least partially responsible for the solar activity cycles (Spiegel and Weiss, 1980; Rosner, 1980; Glatzmaier, 1985; Weiss, 1994; Schou et al., 1998; Zaqarashvili et al., 2010, 2015; Guerrero et al., 2016).

Of all the known concepts playing a noticeable role in understanding of the link between the energy transfer and the darkness of sunspots, let us consider the most significant one, in our opinion. It is based on the Parker-Biermann cooling effect (Parker, 1955a; Biermann, 1941; Parker, 1979b) which explains the high density of the magnetic field suppressing the convective heat transfer from inside the Sun to its surface. It also provides a direct explanation of the lower temperature of the corresponding area on the visible surface of the Sun (see e.g. Fig. 6a and Eq. (31)). According to Fig. 6a and Eq. (28), the cool regions along the entire convection zone, which are formed under the Parker-Biermann cooling effect, have strong magnetic pressure in twisted magnetic tubes. This pressure can be estimated from the condition for the cool umbra (Fig. 9a),

[TABLE]

The anchored magnetic tubes form a pair of sunspots with umbra and penumbra on the solar surface. The neutral buoyancy of the MFT means that when the external density is equal to the internal density ($\rho_{ext}=\rho_{int}$), the temperature inside the tube is lower than that of the ambient medium (see Fig. 6a and Fig. 9a). This leads to the magnetic tube lifting towards the surface of the Sun. So the neutral buoyancy parameters, such as the densities

[TABLE]

the external and internal gas pressures

[TABLE]

plasma $\beta$-parameter (external to internal gas pressure ratio)

[TABLE]

and the toroidal magnetic field in the tachocline

[TABLE]

characterize the virtually empty magnetic tube (see Fig. 9a). $B_{eq}$ is the magnetic field strength in the tachocline at which the magnetic energy density equals to the kinetic energy density of the convective downflows.

Our results confirm (see Eq. (34) for the umbra) that the convective energy transfer is hindered all along the convection zone from the tachocline to the umbra inclusively (see Fig. 9) when the plasma $\beta$-parameter ($\beta=8\pi p_{ext}/B^{2}$) tends to 1, i.e. when the magnetic pressure dominates the internal gas pressure. It is also noteworthy that the convection suppression is more pronounced for the horizontal magnetic field, since the Lorentz field counteracts the vertical magnetic flux in this case. In other words, exceeding certain strength of the magnetic field (see e.g. Eq. (34) and parameters (38)) apparently leads to global convection suppression (Biermann, 1941; Chandrasekhar, 1952; Parker, 1955a; Valyavin et al., 2014; Tremblay et al., 2015).

It is known (see e.g. Miesch (2012); Miesch et al. (2015)) that the Lorentz force feedback leads to significant suppression of the turbulent $\alpha$-effect for a large Reynolds number $R_{m}$, which may be related to the near (e.g. photosphere) magnetic field spirality. The generation of the spiral flows gives birth to the small-scale spirality in the opposite “direction”, which in its turn can suppress the large-scale dynamo, provided it is not dissipated or taken away by the magnetic spirality flux (Miesch, 2012). So, on the one hand, the suppression of the convection from the tachocline to the photosphere is interpreted as the large-scale dynamo suppression (see Fig. 6a and Fig. 10a). On the other hand, with the magnetoconvection in the penumbra (see Eq. (34) for the penumbra), it connects (via the small-scale dynamo action) the sunspots and solar cycle to the large-scale magnetic field of the Sun.

This solution explicitly depends on the lifetime of the magnetic tubes rising from the tachocline to the solar surface. Therefore, because of the magnetic reconnection in the lower layers (see Fig. 4 in Parker (1994)), it is not the final stage of the simulation. The $\Omega$-loop, forming the sunspot umbra (Fig. 10a) through the convection suppression from the tachocline to the photosphere, also gives rise to the convective upflow around the $\Omega$-loop forming the sunspot penumbra. Because of the pre-reconnection (Fig. 10a,b) when the “legs” of the $\Omega$-loop collide (Fig. 10b), the convective flow is generated at the base of the convection zone.

As described by Spruit et al. (1987); Wilson et al. (1990); Parker (1994, 2009), the upward convection flow around the rising $\Omega$-loop brings its “legs” together in such a way that the magnetic field reconnection occurs across this loop. This cuts off the loop from the azimuthal magnetic field, turning it into an O-loop (see Fig. 3 in Spruit et al. (1987) and Fig. 4 in Parker (1994)). After that the azimuthal magnetic field restores its initial configuration and becomes ready for another process with $\Omega$-loop.

Let us now make some important remarks on the turbulent reconnection, the $\Omega$-loop transformation into the O-loop by rapid “legs” closure, the restoration of the initial azimuthal field and the preconditions for another $\Omega$-loop formation in the same place. It is also necessary to explain the physical interpretation of the overshoot process near the tachocline and estimate the velocity $v_{rise}$ and time $\tau_{rise}$ of the magnetic tube rise from the overshoot boundary layer – starting with the azimuthal magnetic flux strength of $B_{tacho}\sim 4\cdot 10^{7}~{}G$ (see Eq. (27)).

One ultimate goal of this section is to determine the general regularities in the theory of MFTs, which are generated by the magnetic buoyancy of virtually empty tubes rising from the tachocline to the surface of the Sun (Fig. 6a and Fig. 9a). Another one is the physical interpretation of the process of MFTs reconnection in the lower layers of the convection zone (Fig. 10). Not only this is related to the magnetic cycles of flux tubes coinciding with the observed Joy’s law for the tilt angle, but both effects (surprisingly enough) are induced by the existence of DM – the solar axions generated in the core of the Sun.

\subsubsubsection

Convective heating and the buoyant rise of magnetic flux tubes in the solar interior: the model of antidynamo flux tubes and the phenomenon of dark matter solar axions

The first problem is devoted to the study of the effect of virtually empty magnetic tubes and the phenomenon of DM of solar axions.

The assumption that the virtually empty magnetic tubes (Fig. 6a) are neutrally buoyant ($\rho_{int}=\rho_{ext}$ (Spruit and van Ballegooijen, 1982)) implies that the temperature inside these tubes is lower than that of the ambient medium (Fig. 6 and Fig. 10a). This leads to the heat inflow, and consequently, the flux tube rises up (see Parker (1975) or Sect. 8.8 in Parker (1979a)). For a horizontal tube with a cross-section of radius $a$ the rise velocity follows from the Parker’s analysis (see Parker (1975), Eq. (60) in van Ballegooijen (1982)):

[TABLE]

where $H_{p}=\Re T_{ext}/g=p_{ext}/g\rho_{ext}=0.08R_{Sun}$ (Böhm-Vitense, 1958; Spruit, 1977; Brun et al., 2011) is the pressure scale height at the tachocline, $T_{ext}$ and $p_{ext}$ are the external gas temperature and pressure, $\delta\equiv Y=\nabla_{e}-\nabla_{ad}=-c_{p}^{-1}dS/d\xi=-c_{p}^{-1}H_{p}dS/dz$ is the dimensionless entropy gradient (see van Ballegooijen (1982); Smolec and Moskalik (2008, 2010)), $\nabla_{e}\equiv d\ln T_{ext}/d\ln p_{ext}$ and $\nabla_{ad}\equiv(\partial\ln T/\partial\ln p)_{s}$ are the local and adiabatic temperature gradients in external and internal plasma (Spruit, 1974; van Ballegooijen, 1982; Christensen-Dalsgaard et al., 1995), $s$ is the specific entropy, $c_{p}$ is the heat capacity at constant pressure, and $\tau_{d}$ is the radiation and/or convection diffusion time of the flux tube:

[TABLE]

where for the fully ionized gas $c_{p}=2.5\Re$ ($\Re$ is the gas constant in the ideal gas law $p=\rho\Re T$), $T(z)$ and $\rho(z)$ are the mean temperature and density; $k_{e}$ is the radiative heat conductivity (see Eq. (36) in van Ballegooijen (1982)); $\ell_{ov}\approx 0.37H_{p}$ (van Ballegooijen, 1982; Christensen-Dalsgaard et al., 2011) is the thickness of the overshoot layer; the total radiative energy flux $F_{tot}=L/(4\pi r^{2})$ depends on the Sun luminosity $L$; $g$ is the gravitational acceleration.

Next we apply the condition of hydrostatic equilibrium, $dp/dz=\rho g$, when the adiabatic temperature gradient $(dT/dz)_{ad}=g/c_{p}$ may be used, and the neutral buoyancy of the flux tube in the overshoot zone $(|\delta T|/T_{ext})^{-1}\sim\beta\equiv 8\pi p_{ext}/B^{2}$. This way we are able to estimate the time of the radiative and/or convective diffusion $\tau_{d}$ (see Eq. (40)) of the flux tube:

[TABLE]

where

[TABLE]

[TABLE]

Here $\nabla_{rad}=(\partial\ln T_{ext}/\partial\ln p_{ext})_{rad}$ is the radiative equilibrium temperature gradient; $(dQ/dt)_{1}$ is the rate of radiative heating, which only depends on the thermodynamic parameters $k_{e}$ and $T_{ext}$ of the ambient plasma, depending only on the radial distance from the Sun center (Spruit, 1974; van Ballegooijen, 1982).

As a result, it is not difficult to show that the van Ballegooijen model combining equations (39)-(43) gives the final expression for the rise time by radiation and/or convective diffusion from the boundary layer of the overshoot to the solar surface,

[TABLE]

and the lifting speed of the MFT from the overshoot boundary layer to the surface of the Sun,

[TABLE]

which are almost identical to the equations (29) and (30) of Fan and Fisher (1996).

Hence, we understand that the van Ballegooijen model is a special case for magnetic fields of $B_{tacho}\leqslant 10^{5}~{}G$, under which a magnetic dynamo can exist. On the other hand, we know that based on the holographic BL mechanism, generating (in contrast to dynamo!) the toroidal magnetic field in the tachocline, the universal model of flux tubes predetermines the existence of not only the fields of $B_{tacho}\leqslant 10^{5}~{}G$, but also the strong magnetic fields of the order $B_{tacho}\sim 10^{7}~{}G$.

Unlike the van Ballegooijen model, we adopt the universal model of MFTs with

[TABLE]

where

[TABLE]

Here $(dQ/dt)_{2}$ represents the radiative diffusion across the flux tube due to the temperature difference ($\delta T\equiv T-T_{ext}$) between the tube and the external plasma (see Fan and Fisher (1996)).

Using simple calculations of equations (39) and (46) for MFTs, it is easy to show that with the help of the total expression

[TABLE]

of the universal model

[TABLE]

and

[TABLE]

which is the general case of the so-called universal model of van Ballegooijen-Fan-Fisher (vanBFF model).

On the other hand, let us remind that on the basis of the BL holographic mechanism, generating the toroidal magnetic field in the tachocline, the universal model of flux tubes is predetermined by the existence of strong magnetic fields of the order of $B_{tacho}\sim 10^{7}~{}G$. Since the physics of the holographic BL mechanism does not involve a magnetic dynamo, we often refer to it as the universal antidynamo vanBFF model. It is determined by the following total energy rate per unit volume:

[TABLE]

[TABLE]

[TABLE]

where we used the relation $|\delta T|/T_{ext}\sim 1/\beta$; the parameter $\alpha_{1}^{2}\approx 5.76$ (Fan and Fisher, 1996; Weber and Fan, 2015); $\vec{F}_{rad}$ is the radiative energy flux (Spruit, 1974); $\nabla_{e}\sim 1.287\nabla_{rad}$ (see Table 2 in Spruit (1974)); $H_{p}/a$ is a factor for the lower convection zone (see van Ballegooijen (1982)).

As a result, we understand that the heating rate of MFTs consists of the rate of radiative and convective heating (see Eqs. (47)-(52); Eq. (10) in Fan and Fisher (1996) and Eq. (7) in Weber and Fan (2015)):

[TABLE]

where $S$ is the entropy per unit mass. The first term $(dQ/dt)_{1}$ determines the mean temperature gradient between the lower convection zone and the overshoot (see Eq. (51)). It deviates from the radiative equilibrium substantially, implying the existence of the nonzero divergence of the heating radiation flux. The second term $(dQ/dt)_{2}$ (see Eq. (52)) represents the radiation diffusion through the flux tube because of the temperature difference between the tube and the external plasma. Its effect is to reduce the temperature difference.

Hence, it is clear that with strong toroidal magnetic fields in the tachocline (of the order $>10^{5}~{}G$) the second term $(dQ/dt)_{2}$ (in contrast to van Ballegooijen (1982); Fan and Fisher (1996); Weber and Fan (2015); Weber and Browning (2016)) is the dominant source of convective heating. For the toroidal magnetic field $B_{tacho}\sim 4\cdot 10^{7}~{}G$ in the tachocline, our estimate of the second term

[TABLE]

is related to the first term $(dQ/dt)_{1}\approx 29.7~{}erg\cdot cm^{-3}\cdot s^{-1}$ (see Eq. (43); see also Eq. (18) in Fan and Fisher (1996)) and the following parameters: $\alpha_{1}^{2}\approx 5.76$ (Fan and Fisher, 1996; Weber and Fan, 2015), $\nabla_{e}\approx 0.4$ (see Table 2 in Spruit (1974); Fan and Fisher (1996)), $1/\beta=B_{tacho}^{2}/8\pi p_{ext}=1$, $\nabla_{e}/\nabla_{rad}\approx 1.287$, $a=(\Phi/\pi B_{tacho})^{1/2}\leqslant 0.1H_{p}$ (see Fan et al. (1993)) with an average value of magnetic flux $\Phi\sim 10^{21}~{}Mx$ (see e.g. Zwaan (1987)).

Taking into account the consequences of the non-local theory of mixing length (Spruit, 1974; Spruit and van Ballegooijen, 1982), it can be shown that thin, neutrally buoyant flux tubes are stable in the stably stratified medium, provided that its field strength $B$ is smaller than a critical value $B_{c}$ (van Ballegooijen, 1982), which is approximately given by

[TABLE]

So, for the maximum value of the toroidal magnetic field ($B_{tacho}\sim 4\cdot 10^{7}~{}G$) the estimate of the rise time of the radiation and/or convection diffusion of the flux tube (see Eq. (48))

[TABLE]

and the lifting speed of an almost empty magnetic tube (see Eqs. (54), (55))

[TABLE]

show that the existence of MFTs on the surface of the Sun is meaningless.

There is one more beautiful problem which is associated with our problem of almost total suppression of radiative heating in virtually empty magnetic tubes (see Fig. 9). Let us remind that photons going from the radiation zone through the horizontal field of the O-loop near the tachocline (see Fig. 9a and Fig. 6a) are turned into axions, thus almost completely eliminating the radiative heating in the virtually empty magnetic tube. Some small photon flux can still pass through the “ring” between the O-loop and the tube walls (see Fig. 6a and Fig. 9a) and reach the penumbra. Let us denote the area of the magnetic tube “ring” by $\pi a_{axion}^{2}$.

The axion-photon interactions may be described by the incoherent Primakoff (1951) process, when the axions are born in e.g. the core of the Sun, or the coherent process in the presence of the external magnetic field, when the axions can be turned into photons and back. This is similar to neutrino oscillations, and the external magnetic field is necessary to compensate the spin violation in the case of photon-axion oscillations (Raffelt and Stodolsky, 1988; Mirizzi et al., 2005; Hochmuth and Sigl, 2007; Fairbairn et al., 2011; Marsh, 2016).

We are interested in the axion-photon oscillations for the photons, coming from the radiation zone near the tachocline and passing through the horizontal magnetic field of the O-loop (see Fig. 11a, Fig. 9a and Fig. 6a). They are intensively converted into axions, thus almost exhausting the radiative heating in the virtually empty magnetic tube.

So, it is clear that the radiative heating coming from the overshoot boundary layer can only pass through the cross-section of the magnetic tube “ring”. As a result, assuming the mean cross-sectional radius of the magnetic tube

[TABLE]

we apply a new analysis of the universal vanBFF model (see Eq. (49) and (47) (analogous to van Ballegooijen (1982)) or (49) and (55) (analogous to Eq. (29) in Fan and Fisher (1996))), where the calculated values such as the magnetic flux $\Phi$ and the rise speed $(v_{rise})_{axion}$ of the MFT to the surface of the Sun, do not contradict the known observational data:

• •

the value of the magnetic flux of the tube

[TABLE]

which is in good agreement with the observational data of Zwaan (1987);

• •

the time of radiation diffusion of the flux tube (48)-(50)

[TABLE]

and, as a consequence, the magnitude of the lifting speed of the MFT (see Eq. (39))

[TABLE]

which are almost identical to the observational data of the known works by Ilonidis et al. (2011, 2012, 2013) and Kosovichev et al. (2016).

In this context, it is known that Ilonidis et al. (2011, 2012, 2013) and Kosovichev et al. (2016) recently detected significant magnetic perturbations at a depth of about 42−75 Mm (i.e. $\sim 0.9R_{Sun}$) and showed that these perturbations were associated with magnetic structures that emerged with an average speed of 0.3−1.4 km/s and appeared at the surface several hours – 2 days after the detection of the perturbations. Interestingly, the results of several attempts to detect emerging magnetic flux prior to its appearance in the photosphere can be compared to our theoretical estimates (60)-(61) and demonstrate surprising agreement as regards the velocity and time of the magnetic tube rise.

Hence, a very important question arises as to how the values of the rise time and speed of the MFTs, which are predetermined by the radius of the magnetic tube (see Eq. (58)), do not contradict the observational data of the Joy’s law. If this is the case, under strong magnetic fields in the tachocline, which are generated by the holographic principle of quantum gravity (see Fig. 4b), DM of solar axions causes the appearance of a “ring” around a MFT. Photons of the near-surface radiation of the overshoot regions passing through the horizontal magnetic field of the O-loop (see Figs. 6, 9 and 11) are converted into axions, thus completely suppressing the radiation heating $(dQ/dt)_{1}$ in some cross-section of the magnetic tube (see (58)).

On the other hand, the most interesting point is that if the axions, which are born in the Sun core, are directly converted into X-rays near the tachocline, then the axion-photon oscillations predetermine the appearance of magnetic sunspot cycles (see Sect. 3.2 and Rusov et al. (2015)).

It means that if the rising speed of the buoyant magnetic tube, which is determined by the values of the magnetic field $\sim 4.1\cdot 10^{7}~{}G$ and the cross-section radius $\sim 3.7\cdot 10^{-4}~{}H_{p}$, causes the appearance of MFTs in the form of sunspots, then the parameters of the universal vanBFF model, such as the magnetic cycles, are almost identical to the observational data of the tilt angle of Joy’s law.

\subsubsubsection

Magnetic reconnection of magnetic tubes in lower layers and the observed features of the tilt angle of Joy’s Law

The problem is devoted to physics of the magnetic reconnection of a magnetic tube in the lower layers, which is associated with the so-called reconnecting dynamo and the observed features of the tilt angle of Joy’s law.

Let us consider the physics of turbulent reconnection in the lower layers of a magnetic tube (see Fig. 10), which, as we discussed above, determines the connection (Fig. 10b) between the global dynamo and the sunspot cycle. According to the prevalent model, the solar dynamo forms thin MFTs with a strength of $\sim 10^{5}~{}G$ near the tachocline (D’Silva, 1993), and some of these tubes can become magnetically buoyant and rise to the surface, forming sunspots and active regions (Parker, 1955b, 1979a; Choudhuri et al., 1995; Fan and Fisher, 1996; Fan, 2009, and Refs. therein). However, until now it has been believed (Jabbari, 2016; Jabbari et al., 2016, and Refs. therein) that numerical and observational studies do not support this scenario. This is due to the fact that the theoretical model of the magnetic field vector tilt in sunspots and bipolar regions (D’Silva, 1993) cannot describe the known observed latitudinal dependence of the magnetic loops tilt (often called Joy’s law (Hale et al., 1919)), where the latitudinal distribution of the mean magnetic field on the surface (and sunspots) as a function of time has the form of a butterfly diagram (see e.g. Fig. 4 in Ossendrijver (2003) and Fig. 17 in Hathaway (2015)).

It is consequently understandable why the physics of virtually empty magnetic tubes (see Fig. 6a, Fig. 9a and Fig. 10a), which depend on a very strong toroidal magnetic field on the basis of the EN effect, does not contradict the possibility of the existence of neutrally buoyant magnetic tubes, emerging from the boundary layer of the overshoot tachocline to the solar surface, as well as the formation of the small-scale dynamo, which is connected to the sunspot cycle (Fig. 9a and Fig. 11a).

Hence, our previous question can be clarified as follows: what is the mechanism of the magnetic reconnection in the lower layers of a neutrally floating tube, which at the beginning (see Fig. 10a) has a practically empty $\Omega$-loop without convection (see Fig. 11a)? In other words, how does the scenario of the model described by well-known physicists (see Spruit et al. (1987); Wilson (1987); Parker (1994, 2009)) transform the turbulent $\Omega$-pumping with the so-called turbulent reconnection (Loureiro et al., 2009; Huang and Bhattacharjee, 2010; Beresnyak, 2017), in which turbulent plasma pumping from the azimuthal field is repeated in the $\Omega$-loop again and again (see Fig. 10 and Fig. 12)?

Below is the brief analysis and summary of the answer. As known, magnetic reconnection is the most striking and amazing observation of the Sun, which causes solar flares. Observations estimate the reconnection speed as small, though not too small, fraction of the Alfvén speed of the so-called rapid reconnection (see e.g. Beresnyak (2017) and Refs. therein). Given the remarkable properties of fast turbulent reconnection, we understand why the mechanism of sunspot formation and corona heating are very strongly associated with the phenomenon of magnetic reconnection (see Loureiro et al. (2009); Huang and Bhattacharjee (2010); Beresnyak (2013, 2017); Jabbari et al. (2016)), which changes the topology of the magnetic field and leads to the conversion of magnetic energy into thermal energy, kinetic energy, and even particle acceleration. Let us discuss the known “unanswered” basic questions related to the nature of dynamo and the origin of sunspots and active regions, together with our respective answers:

$\bullet$ Is there a phenomenon of magnetic reconnection in the lower layers of magnetic tubes? If they appear near the tachocline, then why isn’t there a large-scale dynamo in the virtually empty magnetic tubes?

We have already shown (see Figs. 6a and 9b) that the topological effects of magnetic reconnection inside magnetic tubes near the tachocline form the O-loop (Fig. 13), the “magnetic steps” of which (see $L_{MS}$ in Fig. 6a) participate in the formation of photons of axion origin (see Fig. 6a).

In Fig. 14 one can see the conditions for the magnetic reconnection between the O-loop (green lines) and the unipolar part of the $\Omega$-loop (blue lines) that can organize them in the lower layer (Fig. 14, left) or in the upper layer (Fig. 14, right), thereby showing the appearance of bipolar magnetic tubes in various versions (Fig. 14c,d and g,h).

The O-loop mentioned above may have different directions of magnetic fields, or may not exist within the $\Omega$-loop at all. When there is no O-loop near the tachocline in the $\Omega$-loop, the absence of the horizontal magnetic field of the O-loop (see Figs. 9a and  6a) makes it impossible to convert axions (with average energy $\sim 4.2~{}keV$) into photons (see Fig. 6a). At the same time it passes thermal photons with average energy $\sim 0.95~{}keV$ (in the tachocline (Bailey et al., 2009)), which are strongly scattered near the tachocline (see Fig. 9a). This means that due to the strong scattering of $0.95~{}keV$ photons near the tachocline, already visible photons reaching the photosphere and above generate only “bright” sunspots, i.e. the visible photons form “invisible” sunspots (see Fig. 12), which exist inside the convective zone, but are not observed optically!

Finally, let us note that we have two types of magnetic reconnection inside the $\Omega$-loop near the tachocline, one of which is manifested by the existence of a free O-loop (red lines in Fig. 13, green lines in Figs. 14a and  14e), and the other one practically does not exist. In other words, magnetic reconnection inside the $\Omega$-loop may or may not accidentally give birth to a free O-loop near the tachocline (see Fig. 13). This means that if a free O-loop is created in the $\Omega$-loop near the tachocline, classical sunspots appear on the surface (see black tubes, Fig. 15), and the “transparent” bipolar tubes on the surface appear as “optically invisible” spots otherwise (see white bipolar tubes, Fig. 15).

In this case we are interested in the relationship between cycles of sunspots and a large-scale dynamo. On the one hand, we know that one of the main difficulties with the dynamo of the convective zone is that magnetic buoyancy rather displaces any magnetic flux from the convective zone, without allowing sufficient time to amplify the dynamo (see Parker (1975); Moreno-Insertis (1983); D’Silva and Choudhuri (1993)). On the other hand, we know that according to Choudhuri and Gilman (1987), the magnitude of the effect of magnetic buoyancy is determined by the strength of the magnetic field. This dictates the reduction in density inside the tube compared to its surroundings at a given level. This difference in density $\rho$ can be estimated by considering a magnetic tube in the thermal state, i.e. in complete equilibrium of pressure and temperature with its surroundings (see Choudhuri and Gilman (1987); Choudhuri (1989); D’Silva and Choudhuri (1993)). It is known that if the magnetic field at these levels is supported by a dynamo and the magnetic fields are diffuse (see, for example, Fig. 11a), then the field strength is unlikely to be greater than $\sim 10^{4}~{}G$, since large fields will greatly suppress the dynamo action (Deluca and Gilman, 1986). Since we know that the azimuthal magnetic fields of the tube represent very strong fields near the tachocline ($\sim 10^{7}~{}G$), the strong decrease in internal pressure of the gas, which is predetermined by the thermomagnetic EN effect (see Eq. (25) in Rusov et al. (2015)), yields

[TABLE]

Let us remind that photons coming from the radiation zone through the horizontal magnetic field of the O-loop from the $\Omega$-loop near the tachocline (see Figs. 9a and 6a) are converted into axions, virtually eliminating the radiation heating in the empty MFT (see Fig. 11a). Hence, the most intuitive result of this solution is the fact that the magnetic cycles of sunspots (see Fig. 15) are a consequence of the practically empty flux tubes (see Figs. 9a,  6a and  11a), which, reaching the photosphere, initially do not have the effect of secondary reconnection in the lower or upper layer of the magnetic tube (see Fig. 15). If a large-scale dynamo appears with the spots present, this means that the emerging dynamo with the help of the secondary reconnection near the tachocline (see Figs. 10b,c,  12 and  15) leads to the MFT cut-off (see Fig. 10c) and, as a consequence, the sunspots leaving the surface of the Sun towards the corona.

So, the final connection between the cycles of sunspots and the large-scale dynamo is the fact that the formation of sunspots and their cycles does not depend on the large-scale dynamo, which, strangely enough, is predetermined only by the process of the sunspots liftoff from the surface of the Sun!

$\bullet\bullet$** How do the primary and secondary magnetic reconnection in the lower layers of flux tubes (with magnetic field strength $\sim 10^{7}~{}G$ (see Fig. 12a) and $\sim 10^{5}~{}G$ (see Fig. 12b), respectively) explain the physics and theoretical estimates of the buoyant tubes – the rise time and the speed of rising to the surface of the Sun, and the tendency of the tilt angle of Joy’s law, which does not contradict the known experimental data?**

Based on the MEQ model of Choudhuri and Gilman (1987) and vanBFF MEQ model (see Sect. 10), we showed that the results of the simulation of flux tube trajectories without adiabatic ring flux drag in the super-adiabatic zone (see Fig. 11a) are well represented in Fig. 16a,b (red lines). The most intriguing results of the simulation of flux tube trajectories are a very strong basis not only for understanding the complex physics, but also for understanding the theoretical estimates of buoyant MFTs, like the rise time and speed of rising to the Sun surface (see (60)-(61), and also analogous Fig. 1 and Fig. 5 in Browning et al. (2016)), and the explanation of the tendency of the tilt angle of Joy’s law (see Fig. 16c,d), which do not contradict the known experimental data, for example, the well-known works of Dasi-Espuig et al. (2010, 2013); Ivanov (2012); McClintock and Norton (2013); Pevtsov et al. (2014); Tlatova et al. (2015); Senthamizh Pavai, V. et al. (2015); Baranyi (2015); Wang et al. (2015); Wang (2017).

A brief explanation of the complex physics is as follows. It is known that the mechanical equilibrium (MEQ) of the flux tube is characterized by neutral buoyancy and a balance of curvature and Coriolis forces due to azimuthal flux along the tube. Indeed, an $\Omega$-loop with magnetic field of the order of $10^{7}~{}G$ is an ascending development of the azimuthal stream trapped under the bottom of the convective zone. This is because the sharp reduction of pressure in the lower layers of the considered $\Omega$-loop produces a gas that not only enhances the concentration of the azimuthal flux of magnetic bundles, but also promotes the buoyancy of the magnetic tube. This means that the magnetic buoyancy of the tube is not only associated with azimuthal flux bundles that are in magnetostatic equilibrium along its entire length, but also is a very important consequence of the thermomagnetic EN effect creating strong toroidal flux tubes on the overshoot tachocline. It is not surprising, but this also means that the solar magnetic fields are generated not by the dynamo action in the lower part of the convective zone, but with the help of the thermomagnetic EN effect in the tachocline (see Fig. 4b).

Hence, the question arises as to how and when the large-scale dynamos are born in the flux tubes. It is known that turbulence is the defining feature of magnetized plasma in the cosmic, astrophysical and, of course, solar environments, which are almost always characterized by very large magnetic Reynolds numbers (see e.g. Loureiro and Boldyrev (2017)). In this case the fast dynamo problem is related to the generation of magnetic fields in the limit of the infinite magnetic Reynolds number $Re_{m}\rightarrow\infty$, and has been extensively studied, for example, by Childress and Gilbert (1995). Among the very well known models of turbulent dynamo, we single out a very rare but remarkable idea of the dynamo fluctuation model (see Blackman (1996); Archontis et al. (2003a, b); Baggaley et al. (2009, 2010)) in which the magnetic field is condensed by thin ropes caused by the multiscale model of turbulence (see Baggaley et al. (2009)). This model may be considered as an implementation of the asymptotic limit $Re_{m}\rightarrow\infty$ for the continuous magnetic field, where the magnetic dissipation, which is predetermined by a very sparse, hot plasma and high Alfvén velocity, is strongly localized in small regions of strong field gradients, for example, in the lower layers of MFTs. This means that in the initially homogeneous plasma $\beta_{average}\equiv(p_{int}/p_{mag})\ll 1$, the nonlinear Alfvén compression waves split the material on the energy scale containing the turbulent vortices, with density increase of the order of $1/\beta$, where $p_{int}$ and $p_{mag}$ are the average internal gas and magnetic pressures. It is noteworthy that the balance between the Lorentz work (e.g. in the virtually empty tube) and the Joule dissipation occurs mainly from small regions where strong magnetic flux structures are concentrated (see Archontis et al. (2003a)). Hence, it becomes clear that since the Alfvén velocity associated with the rarefied regions is large, as a consequence, the magnetic dissipation occurs only by reconnection in the lower and/or upper layers of the flux tubes (see Fig. 14), namely with the help of the large-scale dynamo or the so-called reconnection dynamo (reconnecting flux-rope dynamo (Baggaley et al., 2009)).

Here the question arises as to what is the purpose of the so-called reconnecting dynamo. In this context, we briefly consider the injection of the magnetic and turbulent energy at some large scales inside a magnetic tube. As a consequence, the energy density on a large scale is cascaded down to smaller scales through the Kolmogorov inertial range (see Kolmogorov (1941) and e.g. Goldreich and Sridhar (1995)) and, therefore, at different scales leads to turbulent vortices that mostly move in the direction, corresponding to the local magnetic field, than in the perpendicular direction (Goldreich and Sridhar, 1995). It is very important that in the lower part of the cascade there is a dissipation range, where the gradients in the flow are large enough for effective dissipation, in which the only mechanism of magnetic dissipation is the reconnection of magnetic lines carried out in the straightforward manner. Thus, the reconnection more effectively transforms the magnetic energy of the plasma flow into the thermal one, in our case by the dynamo action, which is predetermined by the so-called reconnecting dynamo (see Baggaley et al. (2009)).

This means that, surprisingly, the reconnecting dynamo is never a source of magnetic field generation, except for the magnetic reconnection source, which is a consequence of the mechanism of magnetic dissipation. The intriguing moment of the reconnecting dynamo is the fact that the dynamo never generates magnetic fields and, unlike magnetic diffusion (see Priest and Forbes (2000)), is sensitive only to the nature of magnetic dissipation, the reconnection of which in the final stage releases the heat energy from the flux tube to e.g. the solar corona. This is the goal of forming a large-scale dynamo, which by means of reconnection organizes the volatilization of a free O-loop with the energy of heating (and without sunspots, see Figs. 10d and 12) from the surface of the Sun to the corona. The known experiments suggest that, according to Baggaley et al. (2009, 2010), the theoretical estimates of the probability distribution function of the energy density released in the reconnecting dynamo have a power-law form (see Fig. 27 in Baggaley et al. (2010)), which does not contradict the observed data, for example, heating the solar corona with nanoflares (see e.g. Charbonneau et al. (2001); Benz (2008); Testa et al. (2014)).

Further, for evaluating other experimental data, for example, the Joy’s law slope data (see e.g. Dasi-Espuig et al. (2010); Ivanov (2012); McClintock and Norton (2013); Pevtsov et al. (2014); Tlatova et al. (2015); Senthamizh Pavai, V. et al. (2015); Isik (2015); Karak and Miesch (2017)), we are interested in the second stage of modeling the trajectory of flux tubes (see Fig. 16), when a magnetic tube that reaches the solar surface above the photosphere changes its shape and structure with the topological effect of magnetic reconnection in the lower layers of the magnetic tube, see Figs. 10b and 12. Moreover, we believe that the secondary magnetic reconnection of the flux tubes leads to the real decrease in the magnetic field to $B\sim 10^{5}~{}G$ at $\sim 0.8~{}R_{Sun}$ (see Fig. 18), at which the evolution of the tubes (in thermal equilibrium (TEQ) with the surroundings it is often referred to as magnetic buoyancy (Parker, 1975)) is controlled by the latitudinal pressure gradient in magnetic layers on the overshoot tachocline that allows a balance between nonzero buoyancy force, curvature force and pressure force in the absence of azimuthal flow (see bottom panel in Fig. 2 in Schüssler and Rempel (2002); see also review and Fig. 5b in Fan (2009)), which generally allows a balance of TEQ between four main forces: nonzero buoyancy, magnetic tension, aerodynamic drag, and Coriolis force.

Despite our complicated theoretical calculations of the trajectories (Fig. 16a,b) of the flux tubes without drag of adiabatic flux ring in the superadiabatic convection zone (red lines: based on the MEQ model by Choudhuri and Gilman (1987) and the vanBFF MEQ model (see Sect. 10)) and involving the flux ring resistance in thermal equilibrium (blue lines: based on the TEQ model by Choudhuri and Gilman (1987)), below we will describe the scenario, which explains the simple physics of why the magnetic tubes emerging on the solar surface, can only be at low, and to a lesser extent at middle latitudes (see Figs. 16a,b and 17b).

The first part of the scenario consists in discussing the physics of magnetic flux buoyancy and, as a consequence, estimating the rise speed of an almost empty magnetic tube, $v_{rise}$, based on the vanBFF model (see Sect. 10), which simultaneously coincides with the known expressions for the buoyancy speed, $v_{B}$ (see e.g. Parker (1975); see also Eq. (17) and Fig. 2 in van Ballegooijen and Choudhuri (1988); Eq. (34) in Fan et al. (1993); Eq. (30) in Khaibrakhmanov et al. (2017)):

[TABLE]

where $v_{A}\equiv B/(4\pi\rho_{int})^{1/2}$ is the Alfvén speed of the magnetic field (see e.g. Roberts and Murdin (2001)), $C_{D}\approx 1$ is the drag coefficient, $\rho_{int}$ is the internal density of the gas in the MFT. It is assumed that the magnetic tube is formed inside the disk at an altitude $z/H_{p}\approx 1$.

For the neutral buoyancy condition ($\rho_{int}=\rho_{ext}\approx 0.2~{}g/cm^{3}$; see Figs. 8a and 11a) and the strong toroidal field of the magnetic tube, $B_{tacho}^{Sun}=4.1\cdot 10^{7}~{}G$, as well as the transverse radius of the “thin” ring $a_{axion}\sim 3.7\cdot 10^{-4}~{}H_{p}$ (see Eq. (58)) between the O-loop and the walls of the magnetic tube (see Fig. 9a), it is not difficult to show that the estimate of the Alfvén speed (see Eq. (63))

[TABLE]

allows us to estimate the analytical coincidence of the rise speed $v_{rise}$ (see Eq. (61)) and the magnetic buoyancy speed $v_{B}$:

[TABLE]

at which for such large magnetic fields there is a significant number of rising tubes at all latitudes (see also the red lines in Figs. 17a and 16a,b).

So, for the considered case, the secondary magnetic reconnection of the flux tubes leads to the real decrease in the magnetic field to $B\sim 10^{5}~{}G$ at $\sim 0.8R_{Sun}$ (see Figs. 16a,b and 18a), and thereby reveals the nonzero magnetic buoyancy (see the blue lines in Fig. 16a,b). This means that for the condition of nonzero buoyancy ($\rho_{ext}\approx 0.09~{}g/cm^{3}$) and the toroidal magnetic field of the flux tube, as well as the transverse radius of the “thin” ring $\sigma_{0}\equiv a_{axion}\sim 100~{}km\approx 4.5\times 3.7\cdot 10^{-4}H_{p}$ (see Fig. 16c,d), the estimate

[TABLE]

allows us to estimate the analytical coincidence of the rise speed $v_{rise}$ (see Eq. (61)) and the magnetic buoyancy speed $v_{B}$:

[TABLE]

where we assume that MFTs are formed inside the disk at a height $z\sim H_{p}$ (see e.g. Khaibrakhmanov et al. (2017)). Below we consider the reason for the rising magnetic buoyancy only at low and middle latitudes.

The second part of the scenario, which is based on the remarkable idea of Kichatinov (1991) (see also Fig. 5 in Krivodubskij (2005)), includes the generation of the magnetic field near the bottom of the convective zone (see Fig. 4b for the holographic BL mechanism) and transfer of the toroidal field from the deep layers to the solar surface, where the efficiency of the magnetic buoyancy transfer is predetermined by the participation of two processes: macroscopic turbulent diamagnetism (see Zel’dovich (1957); Rädler (1968a, b); Vainshtein et al. (1980); Vainshtein and Kichatinov (1983); Stix (1989); Kichatinov and Ruediger (1992); Kitchatinov and Nepomnyashchikh (2016)) and rotational $\nabla\rho$-pumping (see Drobyshevski and Yuferev (1974); Vainshtein (1983); Vainshtein and Kichatinov (1983); Stix (1989); Kichatinov (1991); Ossendrijver et al. (2002); Kitchatinov and Olemskoy (2012)), which are also associated with the process of meridional circulation (see e.g. van Ballegooijen (1982); Spruit and van Ballegooijen (1982); Dudorov and Kirillov (1986); van Ballegooijen and Choudhuri (1988); Y.-M. et al. (1991); Choudhuri et al. (1995); Caligari et al. (1995); Nandy and Choudhuri (2002) and also Khaibrakhmanov et al. (2017)).

Let us note some important properties of macroscopic turbulent diamagnetism. It is known that Zel’dovich (1956, 1957) and Spitzer (1956) discovered the diamagnetism of inhomogeneously turbulent conducting liquids, in which the inhomogeneous magnetic field moves as a single whole. In this case the turbulent fluid, for example, with nonuniform effective diffusivity $\eta_{T}\approx(1/3)vl$ (see Fig. 1 in  Kitchatinov and Rüdiger (2008); see also Fig. 18b) behaves like a diamagnetic one and carries the magnetic field with the effective velocity

[TABLE]

where $l$ is the mixing length of turbulent pulsations, and $v=\sqrt{\langle v^{2}\rangle}$ is the root-mean-square velocity of turbulent motion. The minus sign on the right in Eq. (68) shows the meaning of turbulent magnetism: it is not paramagnetic magnetism, so magnetic fields repel from regions with relatively high turbulent intensity. In other words, macroscopic turbulent plasma diamagnetism and, as a consequence, the so-called macroscopic diamagnetic effect (see Rädler (1968a, b)) in the physical sense is the displacement of the averaged magnetic field $B$ from regions with increased intensity of turbulent pulsations to regions with less developed turbulence (Vainshtein et al., 1980; Krause and Raedler, 1980).

However, there is an interesting problem of the diamagnetic process caused by inhomogeneous turbulent intensity with allowance for the total nonlinearities in the magnetic field. This is due to the fact that up to the present time analytical estimates have been obtained only for the limiting cases of weak and strong magnetic fields. For example, at strong magnetic fields of flux tubes, at $B\gg B_{eq}$, the diamagnetic effect becomes almost negligible, in particular, strong magnetic damping of diamagnetism $\sim B^{-3}$ is obtained for super-equipartitions of fields (Kichatinov and Ruediger, 1992), when turbulence is close to two-dimensional (Zel’dovich, 1957). On the other hand, for very weak fields the diamagnetic pumping, which is predetermined by the intensity of turbulence at $B\ll B_{eq}$, is a very effective process (Kichatinov and Ruediger, 1992).

Between the known limiting cases of weak and strong magnetic fields we are interested in the average toroidal magnetic field of a flux tube, that is $B_{tacho}^{Sun}\sim 10^{5}~{}G$, when $B\geqslant B_{eq}$. This is due to the fact that the secondary magnetic reconnection of the flux tubes (see Figs. 14b,d,f,g) leads to the real decrease of the magnetic field to $B\sim 10^{5}~{}G$ at $\sim 0.8~{}R_{Sun}$ (see the blue lines in Fig. 16a,b). This means that the real decrease in the toroidal magnetic field of the flux tube is a consequence of the formation of the reconnecting dynamo (see Baggaley et al. (2009)), as well as the important appearance of two “anti-buoyancy” effects: the downwardly directed turbulent diamagnetic transfer and the rotational effect of the magnetic $\nabla\rho$-pumping (Vainshtein and Kichatinov, 1983; Krivodubskij, 2005).

In this regard, we consider turbulence with quasi-isotropic spectral tensor (Kichatinov, 1987), which is certainly the simplest representation for inhomogeneous turbulence (see Eq. (2.12) in Kichatinov and Ruediger (1992). As a consequence, the information on the spectral properties of turbulence (given by Eqs. (2.12)-(2.16) from Kichatinov and Ruediger (1992)) is sufficient to reduce the expression for the average electromotive force $\varepsilon$ (see Eq. (2.1) in Kichatinov and Ruediger (1992)) to its traditional form, where only integrations over the wave number $k$ and frequency $\omega$ remain. After such shortening it is possible to get (see Kichatinov and Ruediger (1992))

[TABLE]

with the speed of turbulent diamagnetic transfer

[TABLE]

where $q$ stands for the local velocity spectrum, and the rate of rotational magnetic advection caused by the vertical heterogeneity of the fluid density in the convective zone, i.e. the magnetic $\nabla\rho$-pumping effect,

[TABLE]

The effective speeds $v_{dens}$ and $v_{dia}$ are consequences of the non-uniformity of density and of turbulence intensity, respectively, where the latter is attributed to the known diamagnetic pumping.

We are interested in the problem of reconstructing a strong toroidal field $\sim 10^{7}~{}G$ of flux tubes (see Figs. 17a and 18a), which by the reconnecting dynamo (Baggaley et al., 2009) transform regions of the mean magnetic field $\sim 10^{5}~{}G$ in the convective zone (see Figs. 17b and  18a) and, thereby, allow the organization of the amazing balance between the magnetic buoyancy, turbulent diamagnetism, and the rotationally modified $\nabla\rho$-effect. It can be shown that for the parameters $\beta=B/B_{eq}\sim 1$ and $\cos\varphi$ (see Eq. (3.5) in Kichatinov and Ruediger (1992)), the speeds (70) and (71), which depend on the magnetic field $B$ through the kernels $\Re_{dens}(\beta,\varphi)$ and $\Re_{dia}(\beta,\varphi)$ (see Eqs. (3.6) and (3.7) in Kichatinov and Ruediger (1992)), have the following estimates:

[TABLE]

and

[TABLE]

This raises the question of how a certain balance appears between the speeds of magnetic buoyancy (see Eq. (67) and Fig. 17), diamagnetic pumping (see Eq. (72)), and rotating densely stratified $\nabla\rho$-pumping (see Eq. (73) and Fig. 17b),

[TABLE]

In order to consider the balance Eq. (74), it is necessary to apply the widely used approximation of the mixing length (see e.g. Böhm-Vitense (1958); Bradshaw (1974); Gough (1977b, a); Barker et al. (2014); Brandenburg (2016)), which, according to Kichatinov (1991), fully satisfies this goal. This approximation will be understood as the replacement of nonlinear terms along with time derivatives in the equations for fluctuating fields by means of $\tau$-relaxation terms, i.e. instead of equations (3.1) and (3.9) from Kichatinov (1991), we now have the equation of the radial speed of the toroidal field in the convection zone

[TABLE]

[TABLE]

[TABLE]

where $\tau\approx l/(\langle u^{2}\rangle^{\circ})^{1/2}$ is a typical lifetime of a convective eddy; $l$ is the mixing length; $\langle u^{2}\rangle^{\circ}$ is the mean intensity of fluctuating velocities for original turbulence; $\theta$ is the latitude; $e_{r}\nabla\rho/\rho=-e_{r}g/[(\gamma-1)c_{p}T]$ , where $e_{r}$ is the radial unit vector; $T\cong 1.35\cdot 10^{6}~{}K$ is the temperature at $0.8R_{Sun}$; $g=g_{0}(R_{Sun}/r)$ is the gravitational acceleration, where $g_{0}=2.74\cdot 10^{4}~{}cm/s^{2}$ is the surface value; $c_{p}=3.4\cdot 10^{8}~{}cm^{2}s^{-2}K^{-1}$ (fully ionized hydrogen) is the specific heat at constant pressure; $\gamma=5/3$ is the ratio of specific heats $c_{p}/c_{V}$; $3\kappa=\tau\langle u^{2}\rangle^{\circ}$, where $\kappa\equiv\eta_{T}\approx 5\cdot 10^{13}~{}cm^{2}\cdot s^{-1}$ is turbulent diffusivity supplied by the model of non-rotating convection zone (Spruit, 1974; Gough and Weiss, 1976; Stix and Skaley, 1990; Parker, 2009; Karak et al., 2014, see also Fig. 1 in Kitchatinov and Rüdiger (2008)) and the mixing length relation $\langle u^{2}\rangle^{\circ}=-\nabla\Delta Tl^{2}g/(4T)$, where the $\nabla\Delta T$ is superadiabatic temperature gradient; $v_{p}=(1/6)\tau\langle u^{2}\rangle^{\circ}(\nabla\rho/\rho)$ is the velocity of the magnetic field transfer caused by the density gradient (see Eq. (36) in Vainshtein and Kichatinov (1983)); $\hat{\Omega}=Co=2\tau\Omega$ is the Coriolis number (reciprocal of the Rossby number), where $\Omega$ is the rotation speed, $\tau$ is the turnover time, and the functions

[TABLE]

(see $I_{1}$ and $I_{2}$ in Eqs. (3.12) and (3.21) in Kichatinov (1991)) are

[TABLE]

[TABLE]

(see also analogous Eq. (19) and Fig. 2 in Kitchatinov and Nepomnyashchikh (2016)), which describe the rotational effect on turbulent convection.

According to Käpylä, P. J. et al. (2014), the rotational effect on the flow can be measured with the local Coriolis number $\hat{\Omega}=Co=2\tau\Omega$, especially if $\tau$ is estimated on the basis of the theory of mixing length, which predicts values of $\hat{\Omega}$ reaching more than 10 in the deep layers (see e.g. Ossendrijver (2003); Brandenburg and Subramanian (2005); Käpylä (2011); Käpylä et al. (2011)). However, on the other hand, according to Käpylä, P. J. et al. (2014), the question of whether there are such deep layers of solar and stellar convection zones or not is still open.

Below we show that the estimate of the Coriolis number for solar convection in the deep layer should be $\hat{\Omega}\approx 20$. As a consequence of (78)-(80), the following values are assumed for this quantity:

[TABLE]

at which the radial velocity (75) of toroidal field transport changes the sign at the latitude $\theta^{*}=\arccos\sqrt{\varphi_{2}/\varphi_{1}}\cong 41^{\circ}$, being negative (downward) for $\theta>\theta^{*}$ and positive (upward) for $\theta<\theta^{*}$. Using (77), we find that the value of the radial velocity in the convective zone $v_{dens}^{light}$ (see Eqs. (75)-(77)) near low latitudes (e.g. $\theta^{*}=\arccos(0.985)\cong 10^{\circ}$; see also Fig. 17b) almost completely coincides with the value of the speed (73), which was previously calculated with the help of the magnetic field $B$ in the kernel $\Re_{dens}(\beta,\varphi)$.

This means that the balance of the magnetic buoyancy (see Eq. (67) and Fig. 17) and rotating density-stratified $\nabla\rho$-pumping (see Eq. (73) and Fig. 17b) provides both the process of blocking the magnetic buoyancy at high latitudes,

[TABLE]

and the process of lifting MFTs from the base of the convective zone to the solar surface at low latitudes,

[TABLE]

which are simultaneously predetermined by the following rise time value:

[TABLE]

where $z_{0}\approx 0.1~{}H_{p}$ is the length of the magnetic line at $0.8R_{Sun}$ (see the green band in Fig. 18). Here we must remember that magnetic $\nabla\rho$-pumping of plasma from the azimuthal field through the formation of the $\Omega$-loop is repeated again and again (see Figs. 10 and 12; see also Fig. 4 in Parker (2009)). Moreover, our theoretical estimates of the rise time of the $\Omega$-loop are in good agreement with the experimental observations by Gaizauskas et al. (1983), who mention the repeated appearance of new $\Omega$-loops with an interval of 5-8 days, which simultaneously indicates the continuing convective pumping of plasma.

We have shown above that the sunspots – the dark magnetic regions occurring at low latitudes on the surface of the Sun (see Fig. 17) – are indicators of the magnetic field generated not with the help of the dynamo mechanism (this is very important!), which does not exist here, but with the help of the holographic BL mechanism as components of the model of solar antidynamo (see Fig. 7b). Hence, the question arises as to how the sunspots originating from magnetic buoyancy on the surface only at low and, to a lesser extent, middle latitudes do not contradict the known observational data of the tilt angle of Joy’s law (see Fig. 16c,d).

The common notion about solar magnetic fields is that they are created by the dynamo action. Although the solar dynamo theory currently faces many serious difficulties (see e.g. Karak et al. (2014)), there has been no satisfactory alternative theory so far (Cowling, 1981; Gilman, 1986; Choudhuri, 1989). Avoiding an alternative theory, most physicists believe that it is necessary to study more deeply the known physical problems of various aspects of the theory of solar dynamo (see Karak et al. (2014)) before passing a verdict on it.

Conversely, if the question arises whether there is a satisfactory alternative theory against the action of the dynamo, then our short answer is “yes”. We understand that a strong toroidal magnetic field and a tachocline are generated not by the dynamo action, but by the holographic BL mechanism (see Fig. 4b), which is a consequence of the holographic principle of quantum gravity in the Universe and, therefore, in the Sun. This means that the holographic principle predetermines the formation of the fundamental properties of a nearly two-dimensional surface of the tachocline based on the three-dimensional volume of the Sun below the tachocline, and thus the formation of the thermomagnetic EN effect (see Spitzer, 1962, 2006; Rusov et al., 2015, and also Fig. 6a). Hence, it is easy to show that with the help of the thermomagnetic EN effect a simple estimate of the magnitude of magnetic pressure of an ideal gas in the tachocline can indirectly prove that the repulsive toroidal magnetic field of the tachocline precisely “neutralizes” the magnetic field in the Sun core (see also Eq. (27) and Rusov et al. (2015)). This is the main and at the same time fundamental result of the existence of the holographic BL mechanism, which is a consequence of the remarkable properties of the holographic principle of quantum gravity on the Sun.

Hence, it is very interesting that the holographic BL mechanism, which is a satisfactory alternative theory against the action of the dynamo, predetermines not only the generation of sunspots themselves on the surface at low and middle latitudes (see Fig. 17), but their coincidence with the observed slope angle of Joy’s law, where, as a consequence, the average angle of inclination of bipolar sunspots increases with latitude (see Fig. 16c,d). This is due to the fact that strong toroidal magnetic fields in the overshoot tachocline are generated by the holographic BL mechanism (see Fig. 4b), and bipolar magnetic tubes are created by lifting $\Omega$-loops caused by magnetic buoyancy. As a result, these ascending $\Omega$-loops, according to Choudhuri (1989), will be bent (towards the pole; see also Fig. 19) by the Coriolis force, so that they eventually appear on the surface of the Sun with the tilt angle (see Fig. 16).

Using the magnetic field strength near the bottom of the convective zone of the order of $10^{5}~{}G$ at $0.8R_{Sun}$ (see Figs. 17b and 18a), we find that the Coriolis force plays a dominant role, and the MFTs, starting from the bottom at low latitudes, deviate by $0.8R_{Sun}$ (see Figs. 16a,b and 19) and appear on the surface of the Sun at low and middle latitudes, located in the direction of the poles as sunspots. It is obvious that since the time of radiation diffusion of the flux tube (see Fig. 16a) is related to Eq. (84),

[TABLE]

using the equation of the universal model (48),

[TABLE]

it is possible to estimate the radius of the cross-section of the ring of the MFT:

[TABLE]

Here we are interested in the relationship between the speed of magnetic buoyancy, $(v_{B}^{red})_{axion}$, and the radius of the cross-section of the ring, $\sigma_{0}$, which is identical to $a_{axion}$ based on the vanBFF model (see Sect. 10). Using Eq. (23) from Choudhuri and Gilman (1987)

[TABLE]

which connects the radius of the cross-section $\sigma_{0}\equiv a_{axion}$ (see Eq. (51)) with the thermal velocity in dimensionless coordinates (see analogous Table 1 in Choudhuri and Gilman (1987)), it is not difficult to show that for $(\Delta\rho/\rho_{ext})_{0}\sim 10^{-5}$ (or equivalently at $B_{0}=v_{A}(4\pi\rho)^{1/2}\sim 10^{5}~{}G$) the magnetic buoyancy values $(v_{B}^{red})_{axion}/4.3\cdot 10^{4}~{}cm/s\equiv u_{t}^{\prime}\approx 1.7\cdot 10^{-2}$ and $\sigma_{0}\equiv a_{axion}\sim 100~{}km$ remarkably coincide with the corresponding values of Eqs. (67) and (87), respectively.

As a consequence of (87)-(88), from the understanding of physics of the flux tubes trajectory (i.e. the radius of the cross-section $\sigma_{0}\equiv a_{axion}\sim 100~{}km$ and strong magnetic fields $B\sim 10^{5}~{}G$), the question arises: in what way the theoretical calculations of the trajectory of flux tubes with the participation of the drag force (see Fig. 16a,b) obey Joy’s law, when the values of the tilt of flux loops are very close to the observed values ​​(see Fig. 16c,d), and thereby do not contradict the known experimental data?

In the beginning, according to D’Silva and Choudhuri (1993), we provide our calculations of the flux tube trajectories using the dynamic equation of a thin flux tube (Spruit, 1981; Roberts and Webb, 1979; Ferriz-Mas et al., 1989). The basic equation for a non-axisymmetric ring using spherical coordinates was discussed by Choudhuri (1989). Here we use the equation by Choudhuri (1989) (see Eq. (1); also see Eqs. (5)-(9) in D’Silva and Choudhuri (1993)) with an additional term of drag $D_{n}$ (per unit length):

[TABLE]

where $\xi_{n}=r_{n}/R_{Sun}$, $r_{n}$ is the position of the $n^{th}$ Lagrange marker, $\rho_{e}(\xi_{n})$ is the density, and $\sigma_{n}$ is the radius of the cross-section. For the velocity $\vec{v}_{k,n}$ perpendicular to the local tangent at the $n^{th}$ point we have $\vec{v}_{k,n}=\hat{\vec{l}}\times(\vec{v}_{n}\times\hat{\vec{l}})$, where $\vec{v}_{n}$ is the velocity and $\hat{\vec{l}}$ represents the unit vector along the tangent at the $n^{th}$ point. The expression for $\hat{l}$ was given by Choudhuri (1989). Hence, as a result, the term of drag, as shown in Eqs. (5)-(9) in D’Silva and Choudhuri (1993), will increase with decreasing the size of the flux tube and vice versa.

It is very important to note here that the flux tube trajectories always appear in the overshoot tachocline with the help of the strong toroidal magnetic field $\sim 10^{7}~{}G$ (see Figs. 10a,b and  12), but if by means of magnetic reconnection the initial trajectory (see Fig. 10a) switches over to the following trajectory (see Fig. 10b), then it depends on the appearance of the azimuthal magnetic field $\sim 10^{5}~{}G$, whose radial length is equal to the radial length ($z_{0}\approx 0.1~{}H_{p}$ at $0.8R_{Sun}$) of the meridional field in the lower layer near the tachocline (see green bands in Figs. 17b and 18a).

The main result of the theoretical estimation of the flux tube trajectory parameters with $B_{0}\sim 10^{7}~{}G$ is predetermined by the fact that the parameters of the initial trajectory (see red lines in Fig. 16a,b) and the secondary trajectory by magnetic reconnection with $B_{0}\sim 10^{5}~{}G$ (see blue lines in Fig. 16a,b) are parts of one common trajectory of the flux tubes, in which the motion of the flux ring is mainly due to four basic properties:

The existence of anchored flux tubes with $10^{7}~{}G$ in the overshoot tachocline is a consequence of the fundamental properties of the holographic principle of quantum gravity, one of which (unlike the dynamo action!) generates a strong toroidal field in the tachocline with the help of the holographic BL mechanism (see Fig. 4b). 2. 2.

The existence of the magnetic buoyancy of flux tubes on the surface of the Sun is a consequence of the formation of a secondary trajectory by magnetic reconnection with $B_{0}\sim 10^{5}~{}G$ (see blue lines in Fig. 16a,b), at which, on the one hand, the formation of the rotating density-stratified $\nabla\rho$-pumping with $\sim 10^{5}~{}G$ near the tachocline (see green bands in Figs. 17b and  18a) provides the process of blocking magnetic buoyancy at high latitudes, and on the other hand, it predetermines the existence of the dominant Coriolis force in $\nabla\rho$-pumping with $\sim 10^{5}~{}G$, which ultimately generates an arched ascending loop on the surface of the Sun with a tilt in the lower and, to a lesser extent, middle latitudes (see Figs. 16 and  19a). 3. 3.

The basic properties of an almost empty magnetic tube (see Figs. 6a,  9 and  11), characterizing thin radii of the cross-section ($\sigma_{0}\equiv a_{axion}\sim 100~{}km$) and strong magnetic fields $\sim 10^{7}~{}G$, do not contradict the existence of a thin flux tube, since a virtually empty magnetic tube, unlike thin flux tubes, has not only strong magnetic fields, but also a thin radius of the ring between the surface of the thick O-loop and the wall of the tube (see Figs. 9 and 11). It means that it is not the thin flux tubes with $\sim 10^{5}~{}G$ near the tachocline (see green bands in Figs. 17b and 18a) that serve as a basis for the role of the Coriolis force at a certain latitude of the rising tilt in the direction of the active region. Therefore, some cross-section of the tube (see Fig. 16c,d) and the arched ascending loop of the magnetic field lines (see Fig. 19) are allowed. 4. 4.

The averaged theoretical estimates of the magnetic cycle of flux tubes are practically identical to the observational (averaged) data of the tilt angle of Joy’s law (see Fig. 16c,d).

Since the study of the tendencies of the tilt angle of Joy’s law is very important for understanding the evolution of the solar magnetic field, then unlike the heavy calculations of theoretical estimates of the averaged angle tilt of sunspots with increasing latitude, we can summarize our results (see Eqs. (5)-(9) in D’Silva and Choudhuri (1993) and also Eq. (58) for Fig. 16c,d), which can simultaneously be expressed in the form of simple and understandable physics of Joy’s law. It is known that, according to D’Silva and Choudhuri (1993) and Fan et al. (1993), the Coriolis force is proportional to the buoyancy speed $(v_{B})_{axion}$, which can be estimated taking into account the balance between the buoyancy force (left term) and the drag force (right term):

[TABLE]

or

[TABLE]

where in our case the value of the buoyancy speed $(v_{B})_{axion}$ is identical to the known expressions for the rising speed $v_{r}$ of the loop (see D’Silva and Choudhuri (1993); Fan et al. (1993)). Taking into account Eqs. (18) and (19) by Fan et al. (1994),

[TABLE]

we can obtain by means of (91) the following equation:

[TABLE]

where, using the theoretical values of the slope angle $\alpha$, the latitude angle $\theta$, and the radial parameter $\xi=r/R_{Sun}\sim 0.8R_{Sun}$, we can write the simplified equation (see analogous Eq. (93)) in the following form:

[TABLE]

It is very important to understand that, based on the properties of the trajectories of anchored flux tubes with $10^{7}~{}G$, which are a consequence of the holographic BL mechanism (see Fig. 4b), our studies show that the strength of the toroidal magnetic field is $\sim 10^{5}~{}G$ (near the tachocline see the “green” term of the trajectory in Fig. 17b and  18a) and the corresponding radius of the cross-section ($\sigma_{0}\equiv a_{axion}\sim 100~{}km$) appears by means of reconnection of the flux tubes (see Figs. 10b and 12b). This means that the anchored flux tubes with $10^{7}~{}G$ after magnetic reconnection become the flux tubes with $10^{5}~{}G$. So it is not difficult to understand why the Coriolis force (see Eq. (90)) can explain the tilt angle described by Joy’s law (see Fig. 16c,d) and Eq. (94).

Hence, it becomes clear that if all properties of the trajectories of anchored flux tubes with $10^{7}~{}G$ are considered to be a consequence of the fundamental holographic BL mechanism (see Fig. 4b), then this mechanism generates oscillations of the magnetic field in the Sun core (see Gough (1988); Dicke (1988)) (again, not the dynamo action!), which not only are the result of the formation of magnetic cycles, but both effects, not surprisingly, are caused by the existence of DM – solar axions in the core of the Sun. (see Sect. 10).

Let us remind that there is practically no satisfactory alternative theory against the action of the dynamo (see e.g. Cowling (1981); Gilman (1986); Dicke (1978, 1982, 1988); Gough (1980, 1981, 1988); Choudhuri (1989); Karak et al. (2014)). At the same time, the problem of the dynamo action is not yet solved to the end! Although it was believed that “…To sum up, the dynamo theory of the Sun’s magnetic field is subject to a number of unresolved objections, but alternative theories advanced so far are to much greater objections” (Cowling, 1981). In this sense, we still remember the words of Choudhuri (1989): “…It is perhaps fair to say that a major uncertainty remains at the present time regarding the question of where exactly the dynamo process takes place. If we assume the dynamo to operate within the main body of the convection zone, then the magnetic buoyancy poses a serious problem. On the other hand, until we understand how the magnetic flux can get out of the clutches of the Coriolis force, the hypothesis that the dynamo operates at the bottom of the convection zone remains at best a far-fetched speculation. One may even take an extreme point of view and raise the question whether the dynamo theory itself is the correct theory for the generation of solar magnetic fields. There have been some recent claims that the solar cycle may involve oscillations penetrating to the core of the Sun (Gough, 1988). If these claims turn out to be true, then it will be necessary to understand their implications for the dynamo theory. More theoretical and observational work will certainly be needed before the fate of the dynamo theory can be decided”.

From here we understand that the remarkable properties of the holographic BL mechanism are a consequence of not only the fundamental properties of the holographic principle of quantum gravity, but also DM, the existence of the true nature of which is predetermined, not surprisingly, by the conservation law of quantum-gravitational energy in the Universe, and therefore in our Galaxy, and of course in the Sun!

3.2 Coronal heating problem solution by means of photons of axion origin

To estimate the hadron axion-photon coupling constant, we focus on the conversion probability

[TABLE]

where the complete conversion between photons and axions is possible by means of estimating the axion coupling constant to photons.

The $\sim 4100$ T magnetic field in the overshoot tachocline and the Parker-Biermann cooling effect can produce the O-loops with the horizontal magnetic field $B_{MS}\approx B(0.72R_{Sun})\sim 3600$ T stretching for about $L_{MS}\sim 1.28\cdot 10^{4}~{}km$ (see Eq. (9)), and surrounded by virtually zero internal gas pressure of the magnetic tube (see Fig. 6a,b). Since $P_{a\rightarrow\gamma}\sim 1$, we obtain the following parameters of the hadron axion (see Fig. 20).

[TABLE]

The choice of these values is also related to the observed solar luminosity variations in the X-ray band (see (104)). The theoretical estimate and the consequences of such choice are considered below.

Thus, it is shown that the hypothesis about the possibility for the solar axions born in the core of the Sun to be efficiently converted back into $\gamma$-quanta in the magnetic field of the magnetic steps of the O-loop (above the solar overshoot tachocline) is relevant. Here the variations of the magnetic field in the solar tachocline are the direct cause of the converted $\gamma$-quanta intensity variations. The latter in their turn may be the cause of the overall solar luminosity variations known as the active and quiet Sun phases.

It is easy to show that the theoretical estimate for the part of the axion luminosity $L_{a}$ in the total luminosity of the Sun $L_{Sun}$ with respect to (96) is (Andriamonje et al., 2007)

[TABLE]

As opposed to the classic mechanism of the Sun modulation, the axion mechanism is determined by the magnetic tubes rising to the photosphere, and not by the over-photosphere magnetic fields. In this case the solar luminosity modulation is determined by the axion-photon oscillations in the magnetic steps of the O-loop causing the formation and channeling of $\gamma$-quanta inside the almost empty magnetic $\Omega$-tubes (see Fig. 5 and Fig. 6a). When the magnetic tubes cross the photosphere, they “open” (Fig. 6a), and the $\gamma$-quanta are ejected to the photosphere, where their comfortable journey along the magnetic tubes (without absorption and scattering) ends. As the calculations by Zioutas et al. (2009) show, the further destiny of the $\gamma$-quanta in the photosphere may be described by the Compton scattering, which actually agrees with the observed solar spectral shape (Fig. 21b,c).

Considering the above remarks, we believe that it is necessary to calculate the effect of axion-originated photons in active and quiet phases of the Sun on the solar corona. To do this, in Sec. 3.2.1 we first make the preliminary assessment of the magnetic cycles of sunspots using magnetic flux tubes and photons of axion origin on the corona. In Sec. 3.2.2 the theoretical photon spectra of the corona are calculated using inverse Compton scattering (ICS) of isotropic photons on isotropic (moderately relativistic) electrons. In Sec. 23 we discuss whether there is a dark matter chronometer hidden deep in the Sun core.

3.2.1 Preliminary estimation of magnetic cycles of sunspots with the help of photons of axion origin

From the axion mechanism point of view the solar spectra during the active and quiet phases (i.e. during the maximum and minimum solar activity) differ from each other by the smaller or larger part of the Compton spectrum, the latter being produced by the $\gamma$-quanta of axion origin ejected from the magnetic tubes into the photosphere (see Fig. 4 in Chen, Peter, Bingert and Cheung (2015)).

A natural question arises at this point: “What are the real parts of the Compton spectrum of axion origin in the active and quiet phases of the Sun, and do they agree with the experiment?” Let us perform the mentioned estimations being based on the known experimental results by ROSAT/PSPC, where the Sun coronal X-ray spectra and the total luminosity during the minimum and maximum of the solar coronal activity were obtained (Peres et al., 2000).

Apparently, the solar photon spectrum below 10 keV of the active and quiet Sun (Fig. 21a) reconstructed from the accumulated ROSAT/PSPC and ASCA/SIS observations can be described by two Compton spectra for different column densities rather well (Fig. 21b,c). This gives grounds for the assumption that the hard part of the solar spectrum is mainly determined by the axion-photon conversion efficiency:

[TABLE]

where ${d\Phi}/{dE}$ is the observed solar spectra during the active (red line in Fig. 21a,b) and quiet (blue line in Fig. 21a,c) phases, $\left({d\Phi}/{dE}\right)_{CS}$ represents the power-like theoretical spectra of the Compton-scattered photons of axion origin at 0.5 - 2.0 keV,

[TABLE]

where the power-law decay with the “semi-heavy tail” takes place in practice (Lu et al., 1993) instead of the so-called power laws with heavy tails (Lu and Hamilton, 1991; Lu et al., 1993) (see e.g. Figs. 3 and 6 in Uchaikin (2013)). Consequently, the observed corona spectra ($0.5~{}keV<E<2.0~{}keV$) (shaded area in Fig. 21b)

[TABLE]

and (shaded area in Fig. 21c)

[TABLE]

$\left({d\Phi_{\gamma}}/{dE}\right)_{axions}$ is the reconstructed solar photon spectrum fit ($2~{}keV<E<10~{}keV$) built up from three spectra (103) degraded to the Compton scattering for different column densities (see Fig. 21b,c for the active and quiet phases of the Sun, respectively).

As is known, this class of flare models (Eqs. (100) and (101)) is based on the recent paradigm in statistical physics known as self-organized criticality (Bak et al., 1987, 1988; Bak and Chen, 1989; Bak, 1996; Aschwanden, 2011). The basic idea is that the flares are a result of an “avalanche” of small-scale magnetic reconnection events cascading (Lu et al., 1993; Charbonneau et al., 2001; Aschwanden, 2014) through the highly intense coronal magnetic structure (Shibata and Magara, 2011) driven at the critical state by the accidental photospheric movements of its magnetic footprints. Such models thus provide the natural and computationally convenient basis for the study of Parker’s hypothesis of the coronal heating by nanoflares (Parker, 1988).

Another significant fact discriminating the theory from practice, or rather giving a true understanding of the measurements against some theory, should be recalled here (see e.g. Eq. (99); also see Eq. (5) in Lu et al. (1993)). The nature of power laws is related to the strong connection between the consequent events (this applies also to the “catastrophes”, which in turn give rise to a spatial nonlocality related to the appropriate structure of the medium (see page 45 in Uchaikin (2013))). As a result, the “chain reaction”, i.e. the avalanche-like growth of perturbation with more and more resource involved, leads to the heavy-tailed distributions. On the other hand, obviously, none of the natural events may be characterized by the infinite values of the mean and variance. Therefore, the power laws like (99) are approximate and must not hold for the very large arguments. It means that the power-law decay of the probability density rather corresponds to the average asymptotics, and the “semi-heavy tails” must be observed in practice instead.

In this regard we suppose that the application of the power-law distributions with semi-heavy tails leads to a soft attenuation of the observed corona spectra (which are not visible above $E>2~{}keV$), and thus to a close coincidence between the observed solar spectra and $\gamma$-spectra of axion origin (Fig. 21), i.e.

[TABLE]

It means that the physics of the formation and ejection of the $\gamma$-quanta above $2~{}keV$ through the sunspots into corona is not related to the magnetic reconnection theory by e.g. Shibata and Magara (2011) (Fig. 21d), and may be of axion origin.

With this in mind, let us suppose that the part of the differential solar axion flux at the Earth (Andriamonje et al., 2007)

[TABLE]

characterizing the differential $\gamma$-spectrum of axion origin $d\Phi_{\gamma}/dE$ (see $[d\Phi_{\gamma}/dE]_{axions}$ in (98) and (102))

[TABLE]

where the spectral bin width is 6.1 eV (see Fig. 21a); the probability $P_{\gamma}$ describing the relative portion of $\gamma$-quanta (of axion origin) channeling along the magnetic tubes can be defined, according to Peres et al. (2000), from the observed solar luminosity variations in the X-ray band, recorded in ROSAT/PSPC experiments (see Fig. 6 and Table 2 in Peres et al. (2000)): $\left(L_{corona}^{X}\right)_{min}\approx 2.7\cdot 10^{26}~{}erg/s$ at minimum and $\left(L_{corona}^{X}\right)_{max}\approx 4.7\cdot 10^{27}~{}erg/s$ at maximum (see Fig. 21d), and in ASCA/SIS experiments (see Fig. 9 and Table 2 in Peres et al. (2000) and also Figs. 21 and 11b): $\left(L_{corona}^{X}\right)_{min}\approx 2.3\cdot 10^{25}~{}erg/s$ at minimum and $\left(L_{corona}^{X}\right)_{max}\approx 1.5\cdot 10^{27}~{}erg/s$ at maximum (see Fig. 21d).

Consequently, the probability $P_{\gamma}$ describing the relative fraction of the sunspot area on the surface over the magnetic flux tubes during the maximum of Sun luminosity can be determined as:

[TABLE]

and during the minimum of Sun luminosity as:

[TABLE]

where $\langle sunspot~{}area\rangle_{max}\approx 7.5\cdot 10^{9}~{}km^{2}\approx 2470~{}ppm$, $\langle sunspot~{}area\rangle_{min}\approx 7.5\cdot 10^{8}~{}km^{2}\approx 247~{}ppm$ of visible hemisphere is the sunspot area (over the visible hemisphere (Dikpati et al., 2008; Gough, 2010)) for the cycle 22 experimentally observed by the Japanese X-ray telescope Yohkoh (1991) (see Zioutas et al. (2009)); $R_{Sun}=6.96\times 10^{5}~{}km$; $P_{a\rightarrow\gamma}\approx 1$ at maximum of solar luminosity, and $P_{a\rightarrow\gamma}\approx 0.7$ at minimum (see Eqs. (121) and (122)).

The product of the total fraction of axions originating from the Sun core and the fraction of the sunspot area (see Dikpati et al. (2008); Gough (2010)) yields the total fraction of the corona luminosity, or more precisely, the fraction of photon luminosity of axion origin in the corona.

[TABLE]

where $L_{corona}^{X}/L_{Sun}$ is the fraction of the corona luminosity in the total Sun luminosity, $L_{a}/L_{Sun}$ is the fraction of the axion “luminosity”, $L_{Sun}=3.8418\cdot 10^{33}~{}erg/s$ is the solar luminosity (Bahcall and Pinsonneault, 2004).

The axions luminosity fraction $L_{a}$ at the maximum and minimum of the Sun luminosity

[TABLE]

[TABLE]

are determined by the 11-year ADM variations, which are gravitationally captured in the solar interior (see Sect. 23). An important note is that this can lead to the absence of self-annihilation today, which allows large amounts of ADM to accumulate in stars like the Sun (Vincent, Scott and Serenelli, 2015). The ADM particles absorb energy in the hottest, central part of the core, they then travel to a cooler, more peripheral, area before the scattering again and redistribute their energy (Gould and Raffelt, 1990b). This reduces the contrast of temperature across the core region and reduces the central temperature. A colder core produces less neutrinos from the temperature-sensitive fusion reactions, and also less axions from the temperature-sensitive reactions of electron-nucleus collisions (see e.g. Raffelt (1986)). So the fluxes of ${}^{7}Be$ and ${}^{8}B$ neutrino, and axions (see Eqs. (108) and (109)) can be significantly reduced. This is accompanied by a slight change in the $pp$ and $pep$ fluxes, as required by the constancy of the solar luminosity.

Taking into account the known observed variability of the corona luminosity in the X-ray range recorded in the ROSAT/SPC experiments,

[TABLE]

we can compare the “experimental” (on the left; also (110)) and theoretical equations (on the right; also Eqs. (105) - (106) and (108)-(109))

[TABLE]

[TABLE]

which fit well enough!

At the same time, we must keep in mind that the “experimental” data (on the left: Eq. (110)) come from the spectra of the corona, synthesized with the MEKAL spectral code directly from the $EM(T)$ distribution (see Sec. 3.2.1) derived with spectral fitting of ROSAT/PSPC spectra, as reported by Peres et al. (2000) (see Table 2 and thick solid lines in Fig. 6a,b). Therefore, the values of the “experimental” data (on the left: Eq. (110)) are rather rough, while the theoretical data are more accurate.

As a result, in contrast to the variability of corona luminosity (on the left Eqs. (113) and (114)), according to the “experimental” ASCA/SIS data (see Fig. 21d)),

[TABLE]

[TABLE]

the strongest results is the theoretical variability of corona luminosity (on the right: Eqs. (111)-(112)), which practically coincide with the variations of the corona luminosity according to ROSAT/PSPC (on the left: Eqs. (111)-(112)).

The preliminary result is as follows. If the hadronic axions found in the Sun are the same particles that are associated with the known axion coupling to photons (see (96) and Fig. 21a,b,c), it is quite natural that the independent observations of the hard part of the solar photon spectrum are mainly determined by the axion-photon conversion efficiency, and the theoretical estimate for the luminosity fraction from the axions produced in the solar core, the fraction of sunspot area (see Dikpati et al. (2008); Gough (2010)), i.e. from the magnetic cycles (see Eqs. (111)-(112)) of the photons of axion origin.

3.2.2 Asymmetric dark matter and axion from the solar data global fit and coronal heating problem

An interesting problem arises here related to the heating of the practically empty magnetic tube (see Figs. 6a, 11 and 12) and heating of the solar corona, which has long been unresolved (see De Moortel and Browning (2015); Moortel et al. (2016)). There are many assumptions about the unusually high temperature in the corona (see e.g. $T_{average}\approx(1.5-3)\cdot 10^{6}~{}K$ in Güdel and Nazé (2009) and Kariyappa et al. (2011); Cirtain et al. (2013); Peter et al. (2014); Reale (2014); Laming (2015); Peter (2015); Morgan and Taroyan (2017)) compared to the chromosphere and photosphere. It is known that energy comes from the underlying layers, including, in particular, the photosphere and chromosphere. Here are just some of the elements, possibly involved in the heating of the corona: magneto-acoustic and Alfvén waves (see Alfvén and Lindblad (1947); Schatzman (1949, 1962); Parker (1964); Callebaut and Tsintsadze (1994); Jess et al. (2009, 2016); van Ballegooijen et al. (2011, 2014); Arregui (2015); Morton et al. (2016); Laming (2017); Vigeesh et al. (2017)), magnetic reconnections (see e.g. Shibata (1999); Watanabe et al. (2011); Archontis and Hansteen (2014); Xue et al. (2016); Sun et al. (2015); Huang et al. (2018)), nano-flares (see e.g. Aulanier, G. et al. (2013); Testa et al. (2014); Brosius et al. (2014); Cargill et al. (2015); Klimchuk (2015)), Ellerman bombs (see e.g. Watanabe et al. (2011); Vissers et al. (2013, 2015); Libbrecht, Tine et al. (2017)). It is considered (see e.g. Jess et al. (2009)) that the possible mechanism of corona heating is the same as for the chromosphere: convective cells in the form of granulation rising from the depth of the Sun and appearing in the photosphere (see e.g. van Ballegooijen et al. (2011, 2014); Dudík et al. (2014); Cranmer (2015)) lead to a local imbalance in the gas, which leads to the propagation of magneto-acoustic and Alfvén waves (see e.g. Fig. 13 in Laming (2015)) moving in different directions. In this case, a chaotic change in the density, temperature and velocity of the substance in which these waves propagate leads to a change in the speed, frequency and amplitude of the magneto-acoustic waves and can be so large that the gas becomes supersonic. Shock waves appear (see Solanki (2003); Grib and Pushkar’ (2014); Santamaria, I. C. et al. (2016)), which lead to the heating of the gas and, as a consequence, to the heating of the corona.

On the other hand, we believe that the main effective mechanism for heating the solar corona is the emission of axions from the solar core with an energy spectrum with the maximum of about 3 keV and the average of 4.2 keV. These axions are supposed to convert into soft X-rays in very strong transverse magnetic field of an almost empty tube at the base of the convective zone (see Figs. 6a and 11). As a result, X-rays, passing through the photosphere at high speed and scattering in the Compton process, reach the transition region between the chromosphere and corona. The bulk of soft X-rays dissipates in the corona (see Fig. 22c,d).

The total power radiated by X-rays of axion origin is only about one millionth of the total solar luminosity, so there is sufficient energy on the Sun to heat the corona.

It should be recalled here that the energy distribution of the emitted axions is far from being a blackbody spectrum because the spectrum of the incident photons is modulated by the frequency dependence of the cross-section. For the typical solar spectrum, the maximum of the differential flux of axions occurs at $E_{a}/T\approx 3.5$, whereas the average axion energy is $\langle E_{a}/T\rangle\approx 4.4$ (Raffelt, 1986). This means that the average energy of the photon of axion origin can generate a temperature of the order of $T_{a}\sim 1.11\cdot 10^{7}~{}K$ under certain conditions of coronal substances (Priest and Forbes, 2000), which is close to the temperature $T_{core}\sim 1.55\cdot 10^{7}~{}K$ of the Sun core (see e.g. Fiorentini et al. (2001); Fiorentini and Ricci (2002); Bahcall and Pinsonneault (1992); Bahcall et al. (1995)).

This raises an intriguing question about the paradoxical heating of the corona (see e.g. Edlén (1943); Alfvén and Lindblad (1947); Parker (1958); Gibson (1973); Withbroe and Noyes (1977); Parker (1988); Klimchuk (2006); Tomczyk et al. (2007); Aschwanden et al. (2007); Erdélyi and Ballai (2007); Golub and Pasachoff (2009); De Pontieu et al. (2011); Parnell and De Moortel (2012); Reale (2014); Aschwanden et al. (2014, 2015, 2016, 2017); Tan (2014); De Moortel and Browning (2015); Klimchuk (2015); Barnes et al. (2016); Morton et al. (2016)): how do the soft X-rays of axion origin heat the solar corona and flares to a temperature of more than two and, correspondingly, three orders of magnitude higher than in the photosphere?

Below we show why the solar corona is so hot with the help of photons of axion origin.

\subsubsubsection

Solar axion and coronal heating problem solution

It is known (Priest and Forbes, 2000) that traditionally in the atmosphere of the Sun there are three types of eruptions, such as coronal mass ejections, prominence eruptions and eruptive flares, and they are considered bound and are the result of the same physical process. Coronal mass ejections (CMEs) are large-scale mass ejections and magnetic flux from the lower corona to the interplanetary space. It is believed that they should be created by the loss of equilibrium in the structures of the coronal magnetic plasma, which causes sharp changes in the magnetic topology. A typical CME carries approximately $10^{23}~{}Mx$ of flux and $10^{13}~{}kg$ of plasma into space (Priest and Forbes, 2000). During the active phase of the solar cycle, CME can occur more often than once a day. The intermittent appearance of a new magnetic flux from the convective zone (which originates from twist in flux tubes (see Archontis (2012); Schmieder et al. (2014); De Pontieu et al. (2014)) in the corona is the most important process for the dynamic evolution of the coronal magnetic field (Galsgaard et al., 2007; Fang et al., 2010; Archontis and Hood, 2012, 2013), in which the rearrangement of the intersection of closed coronal lines of force causes the accumulation of coronal field stress. When the stress exceeds a certain threshold, the stability of the magnetic field configuration is broken and erupts (see e.g. Fig. 22c; Fig. 2 in Sun et al. (2015)). This model is called a storage model (Yamada et al., 2010), although it is unfortunately known that the question of how magnetic fields rise from the tachocline to the convective zone of the Sun and exit through the photosphere and chromosphere into the corona has not yet been resolved (see e.g. Archontis (2008); Bushby and Archontis (2012); Schmieder et al. (2015)).

Nevertheless, this plausible explanation is associated not only with the appearance of a magnetic flux, but also necessarily with the appearance of photons of axion origin from an empty magnetic tube (see Figs. 6a, 10a, 11a), in which, according to our theoretical and experimental observations (see Fig. 22a), the simultaneous occurrence of a magnetic flux and the flux of photons of axion origin in the outer layers of the Sun is the main mechanism of the formation of sunspots and active regions, being the integral part of the solar cycle, and also of the high energy release in the corona and flares. The physical solution of this problem will be shown below.

First, let us consider the work and method of Peres et al. (2000), who use the X-ray image of the Sun, collected by Yohkoh/SXT, to obtain the corresponding full disk coronal distribution of the emission as a function of temperature – $EM(T)$. From $EM(T)$ the spectrum for wide X-ray spectral range with the corresponding resolution is calculated. In this case, using the summation of the obtained photon spectra of the corona with the available spectral sensitivity of non-solar X-ray observatories, such as ROSAT and ASCA, a synthesized spectra is obtained from the histogram that corresponds to the focal plane of the photon spectrum of the solar corona.

Brief stages of the method of Peres et al. (2000) are as follows. From the Yohkoh/SXT images made using two different filters, the authors get the temperature $(T)$ and the emission measure $(EM)$ in pixels of the images, which then, by sorting out the emission measure values, allows the construction of a histogram of $EM$ vs. $T$.

Let us note that the broadband filters are used which provide reliable thermal diagnostics, since they depend weakly on details of atomic physical models, for example, on the presence of unknown or not very well known spectral lines for the choice of the prevalence of elements (see Reale (2014)). The filter ratio map (see e.g. Fig. 6 in Reale (2014)) provides information on the spatial distribution of temperature and the emission measure (see e.g. Vaiana et al. (1973)). The emission of the optically thin isothermal plasma when measured in the j-th filter passband is

[TABLE]

where $T$ is the temperature and $EM$ is the emission measure defined as

[TABLE]

the distribution of the differential emission measure (see Rosner et al. (1978))

[TABLE]

where $n_{e}$ is the electron density, and $V$ is the volume of the plasma. The ratio $R_{ij}$ of emission in two different filters $i$, $j$ is independent of the density, and depends only on the temperature function:

[TABLE]

The inversion of this relationship gives the temperature value, based on the isothermal assumption.

It should be noted that the calculation of $EM(T)$ differs from $DEM(T)$ mentioned in Peres et al. (2000), because it is the line-of-sight-averaged emission measure distributed throughout the solar corona (but outside the peaks of flares! (see Sect. 5. Summary in Peres et al. (2000)) and differs also from the temperature distribution of the emission measure within one pixel’s line of sight.

Following the steps in the method of Peres et al. (2000), on the basis of the histogram of $EM$ vs. $T$, the emitted spectrum is synthesized using one of the available spectral codes, for example, the MEKAL spectral code (see Mewe et al. (1985, 1986)). Converting the spectrum with device response, and also using the observational consequences (photon noise, distance from the source, exposure time, etc.), the authors get a realistic simulation of the solar corona, observed as if it were a nearby star. Hence, for these spectra, the standard analysis of data taking into account the response of the instrument can be used.

So the “experimental” photon spectra of the corona, synthesized with the MEKAL spectral code directly from the $EM(T)$ distribution (see Fig. 22a), agree with the solar cycle, since, on the one hand, the total coronal luminosity during the solar cycle 22 in the ROSAT/PSPC band ranges from $\sim 2.7\cdot 10^{26}~{}erg\cdot s^{-1}$ at minimum to $\sim 4.7\cdot 10^{27}~{}erg\cdot s^{-1}$ at maximum (see Table 2 in Peres et al. (2000)), and on the other, from the maximum to the minimum of solar activity, the temperature of the maximum emissivity measure (see $EM$ vs. $T$ from Eq. (115)-(116)) shifts from $\sim 2~{}MK$ to $\sim 1~{}MK$ (see Fig. 3 in Peres et al. (2000); Fig. 22c).

To compare the “experimental” photon spectra of the corona (Peres et al., 2000) with the theoretical photon spectra of the corona, we consider the behavior of soft X-ray emissions (SXR) of axion origin (see Fig. 22a) generated inside the magnetic tube (see Figs. 6a, 9a, 11a), which propagate through the convective zone and the photosphere to the surface of the transition region, between the chromosphere and the corona, where the bulk of the X-rays reaches the corona (see Fig. 22c).

The theoretical spectrum of the scattered photons in the corona ($photons$ per $keV$ per $second$ per $electron$) are calculated using inverse Compton scattering (ICS) of isotropic photons on isotropic (mildly relativistic) electrons (see Fig. 3a in Chen and Bastian (2012); Blumenthal and Gould (1970); Tucker (1975); Brunetti (2000); Aharonian and Atoyan (1981); Krucker et al. (2008)), where the energy distributions of electrons with different spectral indices are $\delta=1$ (based on the kinetic energy of the electron, which has a power-law form $\sim(\gamma-1)^{-\delta}$ (Chen and Bastian, 2012). The energy of the incident EUV/SXR photon of axion origin is 4.2 keV (Fig. 22a; see analogous Fig. 5 in Chen and Bastian (2012)) with a photon density of $10^{7}~{}cm^{-3}$, respectively. The limits of the energy spectrum of electrons are $\gamma=10$ ($\approx 4.6~{}MeV$). The results are normalized to one electron above $\sim 0.5~{}MeV$ as was done in Sec. 2.1 of (Chen and Bastian, 2012).

This means that the fraction of the photon spectra of the corona (see Fig. 22a), which are calculated using the inverse Compton scattering (ICS) by means of isotropic photons on isotropic (moderately relativistic) electrons, almost completely coincides with the product of the total fraction of axions ($L_{a}/L_{Sun}$) originating from the core of the Sun, and the fraction of the sunspot area ($P_{\gamma}$).

[TABLE]

[TABLE]

where $(L_{corona}^{X})_{max}\sim 1.04\cdot 10^{28}~{}erg/s$ and $(L_{corona}^{X})_{min}\sim 4.61\cdot 10^{25}~{}erg/s$.

A more accurate explanation is as follows. The Sun as an X-ray star describes solar luminosity at minimum and maximum (see ROSAT/PSPC data in Fig. 21d). Therefore, the fraction $(L_{corona}^{X})_{max}/L_{Sun}$ describing the spectra of photons of axion origin, may be equal to the maximum and minimum of corona luminosity (on the left: Eq. (119)-(120)). On the other hand, the spectra of the photons of axion origin may be calculated from the product of the axion fraction and the sunspot area at the top of magnetic tubes. It is not surprising that the left-hand side of Eqs. (119)-(120) depend only on moderately relativistic electrons, which interact with the photons of axion origin near 4.2 keV, and produce a wide range of photons from the ultraviolet/soft X-rays to $\gamma$-energies, and this is in the corona of the solar disk!

So on the basis of photons of axion origin, we obtain two significant results. First, the solar luminosity of the corona is absolutely identical to the luminosity of the photosphere (which is confirmed by Eqs. (111)-(112) and (119)-(120)), since the number of photons of axion origin, which exit through the magnetic flux tubes of the photosphere, despite the change in their density, is almost equal to the number of photons in the low-density ambient plasma of the corona (see e.g. Fig. 21a). As a result, the experimental ratio $L_{corona}^{X}/L_{Sun}$ (see Fig. 22a) is the same as the theoretical ratio $L_{a}/L_{Sun}\times P_{\gamma}$ (see e.g. Eqs. (111)-(112) and (119)-(120)) in the minimum and maximum solar luminosity.

Second, the energy distribution of the emitted axions is not like a blackbody spectrum, since the spectrum of the incident photons of axion origin is modulated by the frequency dependence of the cross section. As we have already shown, for a typical solar spectrum, the maximum of the axion differential flux occurs at $E_{a}/T\approx 3.5$, while the average axion energy $\langle E_{a}/T\rangle\approx 4.4$ (Raffelt, 1986). This means that the energy of the average photon of axion origin can generate a temperature of the order of $T_{a}\sim 1.11\cdot 10^{7}~{}K$ (see e.g. Fig. 22a,c) under certain conditions of coronal substances, which is close to the temperature $T_{core}\sim 1.55\cdot 10^{7}~{}K$ of the solar core! Under some conditions, the photons of axion origin with very powerful flares in the corona can partially generate energy, which corresponds to a temperature of the order of $T_{a}\leqslant 10^{8}~{}K=10~{}MK$ (see e.g. Fig. 22b,d).

As a result, according to our physical understanding, the coincidence of the theoretical and experimental photon spectra of the corona (see e.g. Eqs. (111)-(112) and (119)-(120)) is connected, on the one hand, with the appearance of the magnetic flux and simultaneously the flux of photons of axion origin in the outer layers of the Sun, and on the other hand, with the manifestation of the basic mechanism of formation of sunspots and active regions correlated with the solar cycle, which determines formation of the corresponding variation in the energy release in the corona and flares.

Thus, the next question is: “What controls the solar cycle? Can axions and/or DM particles control the solar cycle?” If they exist, how are the magnetic cycles of stars around a BH in the center of the galaxy controlled, for example, by modulating the density of ADM particles?

The first two questions seem to us understandable. Below is the idea of using axions and ADM particles to explain the problem of solar cycles.

\subsubsubsection

Is there a chronometer of dark matter hidden deep in the Sun core?

Here we consider the simplified physics of the chronometric model of DM, in which the modulation of the DM density within the solar interior affects the so-called solar modulation of axions. Let us show how the chronometric model of DM predetermines the relationship between helioseismological observables and predictions of solar models, which at the same time correlates with the modulation of magnetic sunspot cycles (see Fig. 21d and 22a) and anticorrelates with the cosmic ray (CR) intensity modulation here in the Solar System and, particularly, in the solar-disk component (see e.g. Figs. 6 and 7 in Ng et al. (2016); see also Fig. 23b).

We already know that the thermomagnetic EN effect (see Sect. 3.1.1) is the basis of the modulation mechanism for the toroidal magnetic field in the tachocline (see Eq. (25)),

[TABLE]

by means of the condition (see Eq. (19)) for the fully ionized plasma at $Z=1$

[TABLE]

If we assume that the magnetic field in the tachocline changes in proportion to the magnetic field on the surface of the solar photosphere, for example, for the maximum and minimum of the 22nd solar cycle (see e.g. Fig. 1 in Dikpati et al. (2008)), then

[TABLE]

[TABLE]

where the average magnetic field is $\langle B_{tacho}\rangle\approx 4.1\cdot 10^{7}~{}G=4100~{}T$ (see Eq. (27)), the magnetic field of the photosphere ($B_{photospere}$) at the maximum is $2800~{}G=0.28~{}T$, and at the minimum it is $2000~{}G=0.2~{}T$ for the 22nd solar cycle (see Pevtsov et al. (2014)). Checking the same formulas for the input data of Bahcall and Pinsonneault (1992), which we now consider as averaged data, gave the density

[TABLE]

[TABLE]

and temperature

[TABLE]

[TABLE]

On the basis of the above-mentioned variations of the magnetic field (see Eqs. (121, 122)), we showed earlier (Sect. 3.2.2 and Fig. 16) the theoretical probability of the temporal variations of solar activity, that correlate with the flux of X-rays of axion origin (see Fig. 10); the latter being controlled by variations of the magnetic field near the overshoot tachocline.

Consequently, let us recall that the axion mechanism of solar luminosity variations, which determines the effect of the “channeling” of X-rays of axion origin along magnetic tubes (see Fig. 6a), allows to explain the effect of practically complete suppression of convective thermal heating and, thus, understand the famous puzzling darkness of sunspots (see e.g. Rempel and Schlichenmaier (2011)), which simultaneously predetermine the physics of variations in solar cycles (see also Joy’s law in Fig. 16c,d).

Hence, it is clear that the source of the cyclic variations of the initial magnetic field and, for example, the axions which are born by electron-nucleus collisions, $\gamma+(e^{-},Ze)\rightarrow(e^{-},Ze)+a$ (see e.g. Raffelt (1986)), should be within the radius of the Sun core. On the other hand, it is known that helioseismology, the diagnostics of which depends on the temperature $(T)$, average molecular mass $(\bar{\mu})$ and their gradients, and consequently, the speed of solar sound as $\delta c_{s}/c_{s}\sim(1/2)[\delta T/T-\delta\bar{\mu}/\bar{\mu}]$ (see Christensen-Dalsgaard (2002); Vincent, Scott and Serenelli (2015)), is unable to measure the temperature or density in this deepest central region below $0.2~{}R_{Sun}$ (see e.g. García et al. (2007); Serenelli (2016)), but can only indirectly estimate these parameters using known solar neutrino experiments. Since neutrinos are formed in several neutrino nuclear reactions of the proton-proton chain and carbon-nitrogen-oxygen cycle (Bethe, 1939; Blanch et al., 1941; Bahcall and Bethe, 1990; Gruzinov and Bahcall, 1998; Haxton et al., 2013; BOREXINO Collaboration, 2014; Haxton, 2014), which take place at different radii of the Sun core (see e.g. Bahcall et al. (2006); Lopes and Silk (2012)), in this case neutrinos, in particular, ${}^{8}B$ neutrino fluxes (see Gando et al. (2012)) can provide an almost precise determination not only of locally averaged temperatures and densities, but also of their possible variations in the core, and as a consequence, of solar cyclic variations in the convective envelope (see e.g. $g$- and $p$-modes in Wolff (2009)).

So after we understood the origin of the first-order flux of solar neutrinos, which was proclaimed a major breakthrough more than a decade ago (see e.g. Akhmedov and Smirnov (2009); Fogli (2015); Smirnov (2016)), we can focus on one of the second-order terms, for example, the variation of solar neutrino fluxes due to the temperature and density variations in the core. Except for several works by Grandpierre (1990, 1996b, 1996a, 2000, 2010, 2015); Grandpierre and ’Goston (2005), such physics is little known (Wolff and Patrone, 2010; Grandpierre, 2015).

In order to find the physical connection between the solar activity and the nuclear processes of the solar core, Attila Grandpierre studied the possible relationship between the local thermonuclear instability and the physical conditions associated with energy and neutrinos produced in the solar core (see e.g. Gough et al. (1996); Morel and Schatzman (1996); Bludman and Kennedy (1996, 1999); Wolff (2009); Wolff and Patrone (2010)). At the same time, the presence of thermonuclear micro-instabilities generated by magnetic instabilities (see Spruit (2002); Braithwaite and Spruit (2017); Brun and Browning (2017)) and, as a result, the formation of hot bubbles in the solar core (Grandpierre, 1990, 1996b, 1996a) causes a significant deviation from the thermal equilibrium and changes the Maxwell-Boltzmann distribution of plasma particles, since the dynamic system is already far from thermodynamic equilibrium (Grandpierre, 2000). Such a complex dynamic Sun ceases to be a closed system, since the production of its energy is partially regulated by tiny external influences, such as planetary tides (see e.g. Javaraiah and Gokhale (1995) and Refs. therein; de Jager and Versteegh (2005); Wilson et al. (2008); Scafetta (2012a, b); Cionco and Compagnucci (2012); Charbonneau (2013); Cionco and Soon (2015, 2017); Liu (2016); Cionco et al. (2018)). Using a non-Maxwellian distribution, for example, the Tsallis distribution, which was applied by means of the Fokker-Planck equation kinetic approach (see Kaniadakis et al. (1996); Kaniadakis (2001); Ribeiro et al. (2017)), such a modification of the system leads to the increase in the temperature of the solar core, which at a given solar luminosity can compensate for non-standard cooling, and thereby, through the generalized Tsallis statistics, maintain the temporal behavior of the neutrino flux (see Kaniadakis et al. (1996); Grandpierre (2004); Grandpierre and ’Goston (2005)) associated with a known estimate of the modulation of the solar neutrino flux in the deep interiors of the Sun (see e.g. Wolff (2009); Khondekar et al. (2012); Sturrock et al. (2016)).

Omitting some of the interesting challenges of Grandpierre (2015) – What is the reason for the mixed core or, in other words, how much excess of temperature is needed for the bubbles to reach the solar shell from the core? What is the strength of the magnetic field in the Sun core? Is there a mass flow in the kernel? – we will consider the fundamental problem of the existence of hot bubbles, not only within the solar core, but in general in the solar-stellar interiors (see e.g. Grandpierre and ’Goston (2005); Grandpierre (2010, 2015); Wolff and O’Donovan (2007); Wolff (2009); Wolff and Patrone (2010)).

First, it is not difficult to show that the gravitational energy of the Solar System, supplied by the planetary tides near the center of the Sun, is very small, and less than $10^{22}~{}erg/s$ (see Table 1 in Grandpierre (2004); p. 18 in Grandpierre (2015)). But the most important, though surprisingly simple, is the fact that the Sun orbital motion is a state of free fall; therefore, aside from very small tidal motions, the associated particle velocities do not vary as a function of position on or within the body of the Sun (see e.g. Fig. 1 in Shirley (2006)). It is necessary to prevent the inappropriate use of rotational motion equations for particle modeling due to orbital rotation (an example can be found in Sect. 2 in de Jager and Versteegh (2005)) and the repetition of future errors of this type (see Shirley (2006)). In this case, the solution of the fundamental physics of the interconnection between the orbital and rotational angular momenta of the Sun (see Shirley (2006, 2017a, 2017b)), i.e. the solution of the principal stumbling block for dynamical spin-orbit coupling hypotheses is considered below.

Second, because the gravitational energy of the Solar System, supplied by the planetary tides near the center of the Sun, is very small, such small heating in the core cannot initiate the mass flow at all, but is dissipated in the form of thermal and Alfvén waves (see Grandpierre and ’Goston (2005)). Hence, we can understand why the famous physicists Wolff and Patrone (2010) incorrectly argue that planetary gravitational forcing can cause a mass flow inside the Sun that could carry fresh hydrogen fuel at deeper levels, including the solar core and, as a consequence, increase the rate of solar nuclear fusion.

The results of Wolff and Patrone (2010) may be sharply changed in the physically powerful and beautiful way if there is a mass flow, for example, in the solar interior, but an invisible mass flow of DM like ADM, that correlates with baryonic matter. They have remarkable properties (see Figs. 23 and 24a) for understanding the evolution of the Sun within the halo of DM of the Milky Way (see e.g. Spergel and Press (1985); Gould (1987, 1990); Gould and Raffelt (1990b, a); Lopes and Silk (2002, 2010); Frandsen and Sarkar (2010); Lopes et al. (2014); Vincent, Scott and Serenelli (2015); Vincent, Serenelli and Scott (2015); Vincent et al. (2016)).

In the current cosmological scenario, we are interested in understanding the evolution of the Sun within the halo of DM. The DM problem is an empirical proof of the reality of physics outside the Standard Model (Bertone et al., 2004). Here we demonstrate that the existence of the weakly interacting ADM in our problem (see the fraction of WIMP-axion CDM in Fig. 20a) is a motivation for experimental observations in which the density of DM is about 5 times greater than the density of visible matter (VM):

[TABLE]

where $\Omega$ as usual denotes the density of the given component with respect to the critical density (see Petraki and Volkas (2013); Zurek (2014)). At the same time, multiple-GeV ADM can explain most anomalies if (and only if – see, for example, Vincent, Scott and Serenelli (2015); Vincent, Serenelli and Scott (2015)) the interaction force between DM and nucleons depends on the momentum exchanged between them.

Unlike weakly interacting massive particles (WIMPs), the motivation for asymmetric dark matter (ADM, see Petraki and Volkas (2013); Zurek (2014)) comes from the baryonic sector of the Standard Model and is based on the initial asymmetry between DM and anti-DM to produce the correct relic abundance. As a consequence, these ADM conditions can lead to the absence of self-annihilation today (see Fig. 24a), which allows large amounts of ADM to accumulate in the center of the Sun due to their capture by its gravitational field (Gould, 1987). In this case, ADM captured during the evolution of the Sun is in the central part of the nuclear-reacting core of the Sun, where the radius of the ADM core is inversely proportional to the square root of the ADM mass (Spergel and Press, 1985). When the weakly interacting DM particles absorb energy in the hottest, central part of the nucleus, they then travel to a cooler, more peripheral area, where before the scattering again and again additionally accumulate energy (Gould and Raffelt, 1990b). This reduces the contrast of temperature across the core region and reduces the central temperature by a few percent (see e.g. Lopes and Silk (2002, 2010, 2012); Lopes et al. (2014)).

It is known that the strong dependence of the rate of nuclear reactions on temperature allows the use of neutrino nuclear reactions, such as $pp$, $pep$, ${}^{7}Be$ and ${}^{8}B$, which occur at various locations in the nuclear region, including the radial temperature distribution in the nucleus. When the weakly interacting DM absorbs energy, it cools the solar core and, of course, lowers the neutrino flux, which is a very sensitive probe of central temperature (see e.g. Lopes and Silk (2012)). On the other hand, helioseismic observations cannot directly determine the temperature inside the Sun core, since no one can determine the gas temperature from knowledge of the sound velocity if the chemical composition of the core is not known (see García et al. (2007)). In this case, neutrino and helioseismic information thus complement each other, especially when the presence of particles such as ADM in the Sun, affecting the heat transfer in the solar interior and, consequently, neutrino fluxes, can simultaneously solve a very difficult “solar composition problem” or otherwise called “solar abundance problem” (see e.g. Villante and Ricci (2010); Villante (2010); Frandsen and Sarkar (2010); Taoso et al. (2010); Cumberbatch et al. (2010); Lopes and Silk (2012); Villante et al. (2014); Vincent, Scott and Serenelli (2015); Vincent et al. (2016); Vincent, Serenelli and Scott (2015); Serenelli (2016)), as well as the surprising problem of the variation of luminosity (see e.g. Sheldon (1969); Spergel and Press (1985); Faulkner and Gilliland (1985); Gilliland (1986); Wilson (1987); Gould and Raffelt (1990b, a); Wolff and O’Donovan (2007); Wolff (2009); Wolff and Patrone (2010); Khondekar et al. (2012)).

Despite the tremendous success in predicting neutrino fluxes and describing the bulk structure of the Sun, the Standard Solar Model (SSM) still cannot reproduce the key observables related to helioseismology. In order to bring the solar simulations into line with these precise observables (see Vinyoles et al. (2017); Serenelli (2016) and Refs. therein), we try the solutions beyond the SSM with the help of e.g. ADM (see e.g. Vincent et al. (2016)).

At first, it became clear that the accurate solar models, which include the capture and transfer of heat from standard spin-dependent or spin-independent ADM, can be constructed to satisfy, for example, the solar radius, age and luminosity (Taoso et al., 2010; Cumberbatch et al., 2010; Lopes and Silk, 2012). Further, the improvement in the quality of the solar model was advanced by Vincent, Scott and Serenelli (2015); Vincent, Serenelli and Scott (2015) who found that a light asymmetric particle with momentum and velocity-dependent interactions could be captured in a sufficiently large amount on the Sun and conduct heat in such a way that helioseismic observables could be brought in full agreement with the data. Although this led to the decrease in the predicted neutrino fluxes, a significant improvement in sound velocity and convective zone depth was sufficient to produce a noticeable improvement over the SSM of 6 standard deviations and a potential solution to the solar composition problem. However, only after the limits of direct detection of recoil spectra for DM particles have been well defined (see the experiments of CRESSTII in Angloher et al. (2016) and CDMSlite in Agnese R. et al. (2016) (SuperCDMS Collaboration)), the best solution is spin-dependent $v^{2}$ scattering with a reference cross-section of $10^{-35}cm^{2}$ (at a reference velocity $v_{0}=220~{}km\cdot s^{-1}$), and a DM mass of about 5 GeV (Vincent et al. (2016)).

Here arises a question of how the deviation of the radial sound speed profile $\delta c_{s}/c_{s}$ in the solar interior depends on the values obtained from helioseismological data for the SSM and the ADM model.

It can be shown that the deviation of the radial sound speed profile $\langle\delta c_{s}/c_{s}\rangle$ from values averaged over the solar cycle (Fig. 24b), determined by the solar minimum (model of ADM with axions in Fig. 24 (see analogous Fig. 6 in Vincent et al. (2016))) and solar maximum (GS98 with axions in Fig. 24a (see analogous Fig. 5 in Vinyoles et al. (2015) and Fig. 3 in Schlattl et al. (1999))), demonstrates a good agreement with the non-standard solar model only.

It is clear from the fits (see e.g. the orange line in Fig. 24a; see also Vincent et al. (2016)) that the obtained nonstandard solar model is good, but not the best, and it may never become the best if the solar abundance problem is not really solved as a consequence of the fundamental physics of the Sun. In this sense, as very deeply and interestingly expressed by Vincent et al. (2016), “It is clear from the fits we present here that the addition of heat transport by DM in the Sun can improve the SSM significantly. The improvement is well beyond what one would naively expect from adding just two degrees of freedom. Even if DM is not the solution, the fact that the true solution can be accurately mimicked by DM would seem to indicate something fundamental about the physical processes behind the solution…”

Hence, it becomes clear that instead of the averaged deviation of the radial sound speed profile $\langle\delta c_{s}/c_{s}\rangle$ over the solar cycle, which is consistent with the simplified nonstandard solar model (Fig. 24b), it is necessary to model the exact variations of the nonstandard solar model, which, including heat capture and transport from standard spin-dependent ADM, can be constructed not just to satisfy the solar radius, age and luminosity (see e.g. Taoso et al. (2010); Cumberbatch et al. (2010); Lopes and Silk (2012)), but by varying the ADM density corresponding to the variations of the solar radius (see Meftah et al. (2015, 2016, 2017); Rozelot et al. (2015, 2016)) and luminosity of the Sun (see Willson and Hudson (1988); Foukal et al. (2006); Ulrich et al. (2010); Chapman et al. (2013)) and, as a consequence, modulations of solar abundances that have low metallicity at the Sun maximum (see e.g. Asplund et al. (2009)) and high metallicity at the Sun minimum (see e.g. Grevesse and Sauval (1998)), against different observables including solar neutrinos, surface helium, the depth of the convective shell, and the sound speed profile (see Vinyoles et al. (2017)).

Surprisingly, the existence of ADM can be a remarkable result of the fact that Ng et al. (2016) (Fig. 23b), who for the first time recently discovered a significant time variation in the gamma radiation flux of the solar disk, believes that anticorrelation with solar activity may be the result of the interaction of ADMs arising from the solar core with cosmic rays in the solar atmosphere.

Curiously enough, we have understood that the fundamental physics of the Sun, as if recalling a forgotten, but physically profound question, raised long ago by R.H. Dicke, “Is there a chronometer hidden deep in the Sun?” (Dicke, 1978, 1979, 1988), suggests the possibility of the existence of ADM (as a result of the energy conservation law with the Galactic frame velocity, density and dispersion) in the Sun core, with which, for example, variations in the ADM density and, as a consequence, variations in the neutrino flux, the solar cycle, solar luminosity, sunspots and other solar activities, seem to be paced by an accurate “clock” inside the Sun.

The question is very simple: If the ADM density variation controls the “clock” inside the Sun, then what is the physics behind this process?

4 How the ADM density variation around the black hole controls the ADM density variation inside the Sun

Let us try to find an answer to the important, yet unconventional question: is there an observable connection between the 11-year variations of ADM density in the solar interior and the periods of S-stars revolution around the SMBH at the center of the Milky Way (see e.g. Genzel et al. (2010); Dokuchaev and Eroshenko (2015); Mapelli and Gualandris (2016))?

In order to answer this question, let us first consider all unexpected and intriguing implications of the 11-year modulations of ADM density in the solar interior and ADM around the BH. We will start with a short overview of our current understanding of the Sun and its mysterious dynamo action, which should be supplemented with the more consistent comparison with other stars. Hopefully this will lead us to the future understanding of the synergy between studies of the stars and the Sun (Thompson, 2014).

Let us recall that the major puzzle about the dynamo is the fact that “for the understanding of the solar dynamo, the Sun is not enough” (Judge, 2014). The attempts to understand this behavior often refer to the combination of the differential rotation, convection and meridional convection flow as the cause of the global field modulation (solar activity cycle) through the magnetic dynamo (see e.g. Rempel (2006)). It is known however that the existing models of solar dynamo suffer from a number of disadvantages. Some of them are highly idealized mathematical or computational models which may explain certain principles but do not correspond to the real behavior of the Sun (see Thompson, 2014). Others may contain some special parameters which correspond to the observed large-scale behavior (like the sunspot number of butterfly diagram), but fail to provide any predictions. As a result, none of them describes the features of the Sun and Sun-like stars correctly, leaving a lot of room for the further grounds and explanations (see e.g. Judge (2014); Thompson (2014); Metcalfe and van Saders (2017); Wargelin et al. (2017)).

It is also known (see e.g. Vinyoles et al. (2015)) that the Standard Model of particles is not a complete theory, and unfortunately it does not determine the existence of the solar cycle. Indeed, some extensions are necessary to address the open questions of fundamental physics like: the nature of DM; matter-antimatter asymmetry in the Universe; origin of neutrino masses; strong CP-violation problem. As noted by Vinyoles et al. (2015), “In order to solve these problems, physics beyond the Standard Model is needed and the existence of new particles or non-standard properties of known particles are generally invoked”, e.g. ADM and axion.

We have already examined the fundamental solutions of the problems related to the existence of DM – ADM, which comes from the baryon sector, and axion – in the solar interior:

1. The problem of dynamo, sunspots and coronal heating. In the beginning, we note the BL mechanism and the holographic BL mechanism as components of our model of solar antidynamo. In contrast to the component of the solar dynamo model (see Fig. 4b), the BL mechanism, which is predetermined by the fundamental holographic principle of quantum gravity and, as a consequence, the formation of the thermomagnetic EN effect in the tachocline (see Spitzer (1962, 2006); Rusov et al. (2015); see also Sect. 3.1.1), emphasizes that this process is associated with the continuous transformation of toroidal magnetic energy into poloidal magnetic energy, but not vice versa. On this basis, the general theory of a buoyant magnetic tube is constructed. The tube is born on the boundary between the tachocline and the overshoot layer. This means that the existence of an anchored flux tube with $10^{7}~{}G$ in the overshoot tachocline is a consequence of the fundamental properties of the holographic principle of quantum gravity, one of which (unlike the known dynamo action!) generates a strong toroidal field in the tachocline by the BL holographic mechanism. Hence, the final problem is a generalized problem of the antidynamo model of the MFTs, where the nature of both effects (the EN effect and holographic BL mechanism), including magnetic cycles, manifests itself precisely in the existence of modulation of ADM and, as a consequence, the solar axion in the Sun interior.

In Sect. 3.2.2 we showed that the supposition of the periodic ADM variations in the solar interior, and consequently the axion flux, became possible because of the theoretical correlation of the time variation of the sunspot activity with the flux of the X-rays of axion origin in magnetic tubes (Fig. 11a) up to the photosphere (see Sect. 3.1) and in the corona (Fig. 22). Though the most remarkable result of the ADM variations in the solar interior is the fact that they can serve as a fundamental solution (see Sect. 3.2) of the coronal heating problem (see Fig. 22) and an intriguing assessment (see the hint of Fig. 24b) of the deviation of radial sound speed profile from the values averaged over the solar cycle, which is easily explained by the solar minimum and maximum, providing a good agreement with non-standard solar models.

The key point here is that the magnetic cycles of sunspots (see Figs. 9, 11a and Fig. 6a) are the consequence of the effect of virtually empty MFTs, reaching the photosphere and having no influence of the lower layer reconnection near the tachocline at the first stage (see Figs. 10a,  11a). Surprisingly, this means that the magnetic sunspot cycle does not depend on any magnetic dynamo! When dynamo does appear (the so-called reconnecting dynamo (see Baggaley et al. (2009) and e.g. Fig. 14)), it cuts off the MFTs at the tachocline via the reconnection, thus leading to the sunspot volatilization from the solar surface (see Fig. 10b,c).

2. A complete theory of the Sun with ADM and axion. Earlier in Sect. 3.2.2 we showed that the deviations of radial sound speed profile (Fig. 24a) from values of Vincent et al. (2016) with ADM (solid line) and Grevesse and Sauval (1998) (GS98) with axions (dotted line) using two solar models correspond to the solar minimum and solar maximum, and the average of them demonstrates the total correspondence of the non-standard solar model with the helioseismic data (see Fig. 24b).

Put simply, when the ADM density at the solar minimum causes the temperature and pressure decrease in the core, the matter density increases, which leads to the changes in several key nuclear reactions rate (see e.g. Sect. 2.1 in Vinyoles et al. (2015)). These changes directly impact the predictable solar neutrino fluxes (see detailed results for all neutrino fluxes in Table 6 in Vinyoles et al. (2015)). The most important fact here is that ${}^{7}Be$- and ${}^{8}B$-neutrino fluxes, which are very sensitive probes of the impacts of DM on the core of the Sun, weaken significantly. Meanwhile, these phenomena are simultaneously accompanied by a slight change in the $pp$ and $pep$ fluxes, in accordance with the requirement of reducing the solar luminosity by $\sim 10^{-3}~{}L_{Sun}$ (Willson and Hudson, 1988; Sofia, 2004; Foukal et al., 2006; Ulrich et al., 2010; Chapman et al., 2013), which is the main result of the solar cycle modulation data on the total solar irradiance (Willson, 2014; Dewitte and Nevens, 2016; Kopp, 2016; Coddington et al., 2016; Dudok de Wit et al., 2017; Lean, 2018; Dudok de Wit et al., 2018).

And vice versa, when the lower ADM density during the solar maximum rises the temperature and pressure in the core, the decrease in matter density is accompanied by the higher fluxes of all sorts of solar neutrinos. The ${}^{7}Be$- and ${}^{8}B$-neutrino fluxes increase significantly together with the corresponding changes in $pp$ and $pep$ fluxes, associated with the solar irradiance growth to $L_{Sun}$ (see also Foukal et al. (2006); Fröhlich (2013); Pelt et al. (2017)).

One of the major indirect evidences of the 11-year variations in ADM density in the solar interior is the fact of almost complete coincidence of the helioseismological observations with the predictions of the new non-standard solar model (see Fig. 24b), which in addition to the spectral determination of solar abundances takes into account the impact on the temperature, density and chemical composition of the solar core.

Let us remind that when the ADM is accumulated near the center of the Sun, it may provide an additional mechanism for the energy transfer from the solar core. The presence of such particles leads to a change in local solar luminosity of about 0.1%, which is accessible for the solar seismic experiments. We use the typical ADM with spin-dependent $\upsilon^{2}$ scattering with a reference cross-section $\sigma_{0}\sim 10^{-35}\div 10^{-37}~{}cm^{2}$ and the DM mass of about $m_{\chi}\sim 10~{}GeV$ (see e.g. Fig. 6 in Vincent et al. (2016), Fig. 4 in Rott et al. (2013) or our Fig. 24a).

As a result, it can be shown that the ADM density variations in the solar interior, which provide feedback to the solar cycle, would also be anticorrelated to the variations of ${}^{7}Be$- and ${}^{8}B$-neutrino fluxes (see analogous values in Table 6 in Vinyoles et al. (2017)), axions and photons of axion origin (see Sect. 3.1.3) and magnetic fields (see thermomagnetic EN effect, Sect. 3.1.1) near the solar core, tachocline or photosphere of the Sun. Despite the small population of DM (less than $\sim$1 DM particle per $10^{10}$ baryons, see e.g. Vincent et al. (2016)), these little corrections to the temperature gradient may have a serious impact on the Sun. They affect the solar structure itself – including the speed of sound $c_{s}(r)$ and the convection zone radius $r_{CZ}$ – and the neutrino fluxes from thermonuclear processes because of their strong dependence on the temperature in the core.

3. ADM modulation and periodic cycles of S-stars at the GC. On basis of the experimental data (see Fig. 25) on the oscillations of the sunspot number, the geomagnetic field Y-component and the global temperature we explicitly demonstrate that their periods coincide with revolution periods of S-stars orbiting a SMBH at the GC of the Milky Way. It is absolutely obvious that such a fine coincidence cannot be random. Then the next quite natural question arises: how do the solar and terrestrial observables “know” about motion of S-stars? And a hypothesis inevitably comes to mind: the “carrier” is none other than DM. More specifically, S-stars can modulate DM flows in our galaxy and, consequently, cause variations of DM space and velocity distributions, in particular, at the Sun and Earth positions. Further, these variations may cause the corresponding variations of the Solar System observables by means of some mechanism, e.g. the interaction of DM particles, which correlate with baryonic matter, with the cores of the Sun and the Earth. Such a probable mechanism is a subject of our Sect. 4.1.2. Here our aim is to stress that the available experimental data indicate the frequency transfer from the center of our galaxy to the Solar System. This fact can serve as an indirect evidence of the proposed hypothesis that DM plays the role of the variations carrier.

Among 19 S-stars (see Table 1 of Gualandris and Merritt (2009)), the most precisely measured properties are those of S2: it is bright (14.2 in the K-band) and has a short orbital period of 16.2 years. According to Boehle et al. (2016), they have been able to track its motion since Keck observations of the GC began in 1995, so that the observations now cover more than one full orbit of this star. In addition, when the new analysis technique is combined with the first complete re-reduction of Keck GC speckle images using speckle holography, it will be possible to track the short-period star S38 (K-band magnitude = 17, orbital period = 19 yr) through the speckle years (see Fig. 25a,b).

On the other hand, it is known that the periods of 19 S-stars at the GC lie in the range $11.5~{}yr\leqslant P_{S-star}\leqslant 400~{}yr$ (see Gillessen et al. (2009); Genzel et al. (2010); Gillessen et al. (2013, 2017)). The orbits of 19 remaining S-stars do not lie in the disc: they are consistent with a random distribution in space (see Fig. 25a). The period uncertainties for $11.5~{}yr\leqslant P_{S-star}\leqslant 60~{}yr$ are rather low (Gillessen et al., 2009) because of the high measurement precision of Keck observations of the GC that began in 1995 (see e.g. Fig. 5 in Boehle et al. (2016)) and number $\sim 22$ year points to 2017. This means that the estimates for 13 of 19 S-stars at the GC give the accurate period values for $\sim$70% of S-stars. They can also serve as an evidence for the nonrandom coincidence of these periods with the cycles of e.g. sunspots, geomagnetic field and temperature on the Earth (see Fig. 25b,c). The longer orbital periods, $60~{}yr\leqslant P_{S-star}\leqslant 400~{}yr$ (see Gillessen et al. (2009); Genzel et al. (2010); Gillessen et al. (2013, 2017)), which currently have high uncertainties, may be improved to 2030-2050 thanks to the higher number of yearly points measured for $\sim$95% of S-stars.

Let us consider the question of how the ADM density variations, and ADM luminosity variations $L_{\chi}$, work as a “clock” regulating the tempo of the solar cycle. As we understand, Dikce’s clock (Sect. 23) is a consequence of the ADM density modulation, which correlates with the baryonic matter, in the Sun and in S-stars (Ghez et al., 2008; Gillessen et al., 2009, 2013) orbiting the SMBH (of the radius $\leqslant 0.0024pc$, see e.g. Fig. 25a and Gillessen et al. (2017)) in the Milky Way center ($\approx 4\cdot 10^{6}M_{Sun}$) (see Genzel et al. (2010)), where the capture and annihilation of ADM occur in punctuated stages, clearly correlated with the orbital period (see Scott et al. (2009)). The S-star cluster lying between the BH and the clockwise disk, is one of the most mysterious components of the GC: most of the S-stars are early-type stars and could not form in situ, with a pericenter so close to the SMBH (this is the so-called “paradox of youth” (Ghez et al., 2003)). Among all S-stars ($\leqslant 30$ (Mapelli and Gualandris, 2016)) the closest one is S102, which has a period of only 11.5 years (Meyer et al., 2012).

A striking coincidence of revolution periods of S-stars orbiting a SMBH at the GC of the Milky Way and oscillation periods of such solar and terrestrial observables as the sunspot number, the geomagnetic field Y-component and the global temperature is established on the basis of the corresponding experimental data. Rejecting randomness of this discovered coincidence, we put forward a hypothesis that modulation of DM flows in the Milky Way by the S-stars is responsible for such a frequency transfer from the GC to the Solar System.

Let us estimate the modulations of the ADM around the BH and their possible experimental observations.

4.1 Active galactic nuclei variability as a trigger of the matter accretion onto a supermassive black hole

One of the signature properties of AGNs is their variability. AGNs are the most powerful sources of constant light in the Universe (with the bolometric luminosities as high as $10^{48}~{}erg/s$ or $10^{14}L_{Sun}$ (see e.g. Kozłowski (2016b) and Refs. therein)). Their variability during the periods of months to several years about 10% of the total light must be enormous. There are numerous evidences that AGNs are fed from the matter accretion onto SMBHs (Salpeter (1964); Lynden-Bell (1969); Shakura and Sunyaev (1973); Soltan (1982); Rees (1984); Richstone et al. (1998); Silk and Rees (1998)). The common picture of an AGN is that the material is pulled by the supermassive ($\sim 4\cdot 10^{6}$ solar masses) BH in the center of a galaxy. Due to angular momentum conservation, this material forms the accretion disc around the BH. Some part of the material streams towards the BH boundary and accretes, and another part is accelerated by strong magnetic fields and forms the relativistic jet perpendicular to the disc (Fig. 26a).

For example, the chromatic microlensing of the strongly gravitationally lensed AGN yields the sizes ($\sim$1 AU in the optical band) and temperature profiles of the accretion disks (Czerny et al., 1994; Kochanek, 2004; Dai et al., 2010; Morgan et al., 2010; Blackburne et al., 2011, 2014). The masses of SMBHs in the local Universe are known to be related to some properties of their home galaxies (Magorrian et al., 1998; Marconi and Hunt, 2003; McConnell and Ma, 2013; Kormendy and Ho, 2013; Läsker et al., 2014). The physical mechanisms responsible for such intense accretion episodes in galactic nuclei and their relation to the cosmological evolution of the galaxies are still unknown.

Since the discovery of AGNs numerous studies of their variability appeared, both theoretical and observational. Now it is known that AGNs are the variable sources on all wavelengths (see e.g. Kozłowski (2017) and Refs. therein), and the accretion disk instability simulation (Kawaguchi et al., 1998) demonstrates the best match to the observations in the optical band (Chen and Taam, 1995; Vanden Berk et al., 2004; Kozłowski, 2016b). The exact process driving the variability is unknown though. The typical AGN variability has the stochastic nature (see e.g. Kelly et al. (2009); Andrae et al. (2013); Zu et al. (2013); Kozłowski (2016b, a)). It shows the flat spectrum in the low frequencies (white noise, PSD $\propto\nu^{0}$) and the red noise (PSD $\propto\nu^{-2}$) or even steeper dependence (PSD $\propto\nu^{-3}$) in the high frequencies (see e.g. Mushotzky et al. (2011); Kasliwal et al. (2015); Simm et al. (2016)).

An interesting semi-analytic model by Fanidakis et al. (2011), which shows the formation and evolution of the galaxies in a CDM Universe, seems to be most plausible. According to Fanidakis et al. (2011), the BH and galaxy formation models are coupled: during the evolution of the host galaxy, the hot and cold gas is added to the SMBH by the flows from the cooling gas halo, the disk instabilities and galaxy mergers. This provides the mass and spin of the BH, and the resulting accretion power regulates the consequent gas cooling and star formation. Next, the accretion flow is believed to form the geometrically thin cool disk if the accretion rate is higher than $0.01\dot{M}_{Edd}$, and geometrically thick, radiation-inefficient hot disk if the accretion rate is lower than that. The origin of such dichotomy is yet unknown, but many studies suggest to make the first step towards the explanation why some AGNs fire prominent jets while others do not, and eventually to the explanation of the nature of central driver of the accreting BH (see e.g. Falcke et al. (2004); Schawinski et al. (2015); Oh et al. (2015)). Understanding of the connection between galaxies with AGN, powered by matter accretion onto SMBHs at the centers of galaxies (Silk and Rees, 1998), and DM halo they reside in, is the key to the processes of initiation and feeding of the BHs (see Fanidakis et al. (2013); Leauthaud et al. (2015)).

One of the most advanced ideas of the AGN initialization on the basis of galaxy interactions and disk instabilities is described in the paper by Gatti et al. (2016). The ultimate goal is to single out the key properties of the clustering in two modes, which can serve as the reliable probes for the dominating mechanism of SMBH feeding.

Among the remarkable results of Gatti et al. (2016) we would pick the most intriguing one. Their analysis suggests the presence of both a mild luminosity and a more consistent redshift dependence in the AGN clustering, with AGNs inhabiting progressively less massive DM halos as the redshift increases. In other words, less luminous AGNs are biased towards lower DM halo masses. At high redshift, the average halo mass sensibly moves towards lower values.

This means that these AGNs fed by the matter accretion onto the SMBHs, are observed with high redshift as quasars among the most luminous objects in the Universe. The brightest infrared galaxies really possess both the strong AGN activity and the star formation, which suggests these two phenomena are closely related. The details of whether AGN feeding and outflows drive star formation or vice versa remain unclear, and remain to be elucidated by future observations, especially at far-infrared and X-ray frequencies (see e.g. Silk (2007)).

From here we derive our main assumption that the luminous AGN modulation, related to periodic variability, is determined by the modulation of matter accretion by the SMBH. The assumption about the modulation of baryon accretion is directly connected to the modulation of the DM density in the SMBH.

The Ockham’s razor will strengthen such scenario if the DM density modulation succeeds in solving the problems related to the known observations. We discuss some of them below.

4.1.1 AdS/CFT correspondence and the thermomagnetic Ettingshausen-Nernst effect

In theoretical physics the anti-de Sitter – conformal field theory correspondence, sometimes referred to as “the Maldacena duality” or “the gauge/gravity duality” (see Maldacena (1999)), represents a hypothetical connection between two different types of physical theories – the gravitation theory and the quantum field theory. The AdS spaces used in the quantum gravitation theory formulated in terms of the string theory or M-theory (Becker et al., 2008) are on the one hand. The conformal field theories (CFTs), which are the quantum field theories including those similar to the Yang-Mills theories (see e.g. ’t Hooft and Veltman (1972); ’t Hooft (2005)) describing the elementary particles, are on the other hand. Such equivalence is an example of the holographic duality – a representation of the 3D space on the 2D photographic emulsion (see ’t Hooft (1993); Susskind (1995); Hanada et al. (2014)). Thus, although these two types of theories seem to be very different, they are mathematically identical (Maldacena, 1999; Gubser et al., 1998; Witten, 1998)).

In their ingenious paper Hartnoll et al. (2007) turn to the physics of a BH in the 3D AdS space, which carries both the electric and magnetic charges. This lets them derive the transport equations directly from the quantum field theory. They showed that these theoretical results apparently correspond to the known features of the thermomagnetic EN effect – the perpendicular heat and charge fluxes in the presence of the magnetic field (see Spitzer (1962, 2006); Wang et al. (2006); Hartnoll et al. (2007); Rusov et al. (2015)).

This is however darkened by the fact that the AdS/CFT has been proven to work in certain special cases only, such as the important theory of the EN effect near the quantum phase transitions in the condensed matter, or the dyonic BHs (see Hartnoll et al. (2007, 2016)). This correspondence must be applicable to the more general cases as well, but one cannot show this explicitly so far. The progress in this direction is also often complicated by the absence of the full control over both sides of this duality (see Lee (2012, 2014, 2016)).

Let us recall that the basic link between the field theories on the surface and the gravitation theories in the volume is established via the renormalization group (RG), where the volume variables represent the scale-dependent couplings and the radial direction represents the length scale (see Skenderis (2002); Papadimitriou (2016)). The AdS/CFT correspondence maps the N-dimensional field theory into the (N+1)-dimensional gravitation theory. The extra dimension is usually accepted as a length scale. The conversion of the theories from one scale to another is exactly the thing that the RG does (more precisely, the quantum RG and holography). That is why it is believed to be the key to any proof of the AdS/CFT hypothesis (see Lee (2014, 2016)).

It is also understood that any version of the “holographic renormalization”, trying to bridge the Planck-scale quantum gravitation theories with the observations, must be associated with the minimum measurable length or the maximum measurable momentum. These are predicted by various quantum gravitation theories (such as the string theory or double special relativity) as well as the physics of BHs. The search of these limits is related to the so-called generalized uncertainty principle (GUP) or modified commutation relations between the coordinate and momentum (see Amati et al. (1989); Maggiore (1993); Magueijo and Smolin (2005); Das and Vagenas (2008)). The only GUP consistent with the symmetries and index structure of the modified commutator between the coordinate $x$ and momentum $p$, providing $[x_{i},x_{j}]=0=[p_{i},p_{j}]$ (Jacobi identity) at the same time, is (Ali et al., 2009, 2011):

[TABLE]

[TABLE]

This yields the minimum measurable length and the maximum measurable momentum

[TABLE]

[TABLE]

where $\alpha_{0}$ is a dimensionless number, $\alpha=\alpha_{0}/M_{Pl}c=\alpha_{0}l_{Pl}/\hbar$, $M_{Pl}\approx 22~{}\mu g$ (Planck mass), $l_{Pl}\equiv\sqrt{\hbar G/c^{3}}=1.6\cdot 10^{-35}~{}m$ (Planck length), and $M_{Pl}c^{2}=1.2\cdot 10^{19}~{}GeV$ (Planck energy). According to this approach, the momentum must be modified as (Ali et al., 2009, 2011; Moussa, 2014)

[TABLE]

If one assumes that the parameter $\alpha_{0}$ is equal to unity, the terms with $\alpha$ are significant for the energies (momenta) of the Planck energy order and the lengths of the Planck length order (see Ali et al. (2009)).

It was recently shown (see Das and Vagenas (2008); Ali et al. (2009)) that the GUP leads to the corrections to the Schrödinger and Dirac equations affecting both the relativistic and nonrelativistic quantum Hamiltonians. Ali et al. (2011); Das and Mann (2011) also determined the bound on $\alpha$ demonstrating that the GUP may be detected in the low-energy systems like the Landau levels, Lamb shift, quantum Hall effect and the anomalous magnetic moment of the muon. I.e. the GUP is also applicable to the low energy scales. The most important consequence of this bound is that the space must be discrete, and all measurable lengths must be quantized with the fundamental measurable length of $\alpha\hbar=\alpha_{0}l_{p}$ which cannot exceed the scale of the electroweak interaction (Ali et al., 2011; Das and Mann, 2011; Das et al., 2010).

This rises an interesting problem of the GUP quantum gravity corrections to various quantum phenomena, in particular, to the thermomagnetic EN effect in the tachocline at low energies.

\subsubsubsection

Quantum gravity and the generalized thermomagnetic Ettingshausen-Nernst effect in tachoclines of the Sun and magnetic white dwarfs

Let us consider some solutions of the problem of GUP quantum gravity, emerging in response to the question: “What kind of the mysterious nature of the solar tachocline gives birth to the repulsive magnetic field through the thermomagnetic EN effect?”

According to Ali et al. (2009, 2011), it can be shown that

[TABLE]

where $x_{0i}$, $p_{0j}$ satisfy the canonical commutation relations $[x_{0i},p_{0j}]=i\hbar\delta_{ij}$, and $p_{0}=-i\hbar\partial/\partial x_{0}$, satisfy Eq. (128).

Through Eq. (133) it was shown in Das and Vagenas (2008); Ali et al. (2009); Das and Vagenas (2009) that any nonrelativistic Hamiltonian of the form $H=p^{2}/2m+V(r)$ may be given as

[TABLE]

and the modified Schrödinger equation is

[TABLE]

Let us consider the terms with $\alpha$ and $\alpha^{2}$ as the perturbations, although this higher order Schrödinger equation has the non-perturbative solutions of the form $\psi\sim e^{ix/2\alpha\hbar}$, which may lead to the interesting physical consequences (Ali et al., 2009). Some phenomenological consequences of the modified Hamiltonian of such GUP model were addressed in Das and Vagenas (2009); Das et al. (2010).

So, we examined the Planck scale effects in some low energy system via the GUP, which is a clear prediction of the quantum gravity theories, and found the effect to be small yet nonzero for the fundamental quantum flux in the thermomagnetic EN effect in the tachocline.

The thermomagnetic EN effect in the tachocline of the Sun. Let us note that using Eqs. (128)-(129) and (133), we applied this GUP version to the problem of the thermomagnetic EN effect, which is actually an almost perfect analogy of the superconductivity problem (see Sect. 1 in Das and Mann (2011)). Making use of this analogy, we involved a Schrödinger current (from Eq. (135)) through the density of states of the Landau levels and the substitution $2e\rightarrow e$, since the charge carriers are electrons in this case, and not the Cooper pairs. These effects have not been observed so far, but it is possible to estimate the upper bound on the GUP parameter

[TABLE]

assuming the experimental error of 1 per $10^{n}$, e.g. for $n=2$, $B\approx 4.1\cdot 10^{3}~{}T\approx 8\cdot 10^{-13}~{}GeV^{2}$, $L_{tacho}\approx 4\cdot 10^{4}~{}km\approx 2\cdot 10^{23}~{}GeV^{-1}$ (see Rusov et al. (2015)). The total bound for $l_{n}\geqslant(\Delta x)_{min}=\alpha_{0}l_{Pl}$ (see Eq. (130)) reads:

[TABLE]

It should be emphasized that the obtained upper bound on the GUP parameter is more strict than those obtained in recent experiments comparing the nonlocal relativistic effective field theory with the 8TeV LHC data (see $l\leqslant 10^{-19}~{}m$ in Biswas and Okada (2015)), as well as the low energy macroscopic optomechanical oscillators (see $l\sim 10^{-18}~{}m$ in Marin et al. (2013); $l\sim 10^{-22}\div 10^{-26}~{}m$ in Belenchia et al. (2017); $l\sim 10^{-29}~{}m$ in Bawaj et al. (2015)). This bound is smaller than the electroweak interaction scale, which may indicate the existence of the intermediate length scale between the electroweak and Planck scales. Although the figures are 7 orders of magnitude larger than $l_{Pl}=1.6\cdot 10^{-35}~{}m$, they can serve as a sign of the quantum gravity corrections to various quantum phenomena including the thermomagnetic EN effect in the tachocline.

The generalized thermomagnetic EN effect by analogy to the solar tachocline in the magnetic white dwarfs: the stably burning hydrogen shell on the helium core. Now we understand that it is not “some kind of the mysterious nature” that gives birth to the thermomagnetic EN effect and, consequently, to the repulsive magnetic field in the solar tachocline (see Eq. (27)). It is the quantum gravity that does. The question is how exactly does the quantum gravity induce the repulsive magnetic field?

As it was noted above, the fundamental minimum length scale cannot exceed that of the electroweak interactions. Therefore GUP has to modify the density of states, which impacts the statistical and thermodynamic properties of the ideal fermion and boson quantum systems. According to Ali et al. (2011), the number of quantum states in a certain volume of the momentum space should change in the following way:

[TABLE]

Leaving aside the complex calculations of the modified density of states (see Camacho (2006); Moussa (2014)), related to the distribution function, the number of particles, the total energy and pressure of the gas, it is important to understand that quantum gravity induces repulsive forces between fermions and bosons in their ground state, thus increasing the pressure in quantum systems.

Consequently, while increasing pressure in the ground state, quantum gravity induces repulsive magnetic fields e.g. in the solar tachocline (see Eqs. (26)-(27)).

It is interesting that the quantum gravitation behaves differently if the fermions are in the degenerate state (Camacho, 2006; Moussa, 2014). In this case the repulsive force is smaller, and the pressure is smaller too. The smaller pressure of the ground state of fermions implies the smaller radius-to-mass ratio in white dwarfs as a test for GUP. For example, according to Camacho (2006); Ali and Tawfik (2013); Moussa (2014), the radius contraction is proportional to the mass of the white dwarf and is divergent with the mass when it reaches the Chandrasekhar limit. This result is consistent with the current observations indicating that the white dwarfs have smaller radii that it was theoretically predicted (Provencal et al., 2002; Mathews et al., 2006).

The most interesting phenomenon is, of course, the generalized thermomagnetic EN effect in the tachoclines of stars, e.g. the magnetic white dwarfs (see Valyavin et al. (2014)). Let us start approaching this problem by estimating the magnetic field in the tachocline. In the quasi-steady magnetic field the thermomagnetic current generated in the fully ionized hydrogen plasma is (see Eqs. (4-1) in Spitzer (1962)):

[TABLE]

where

[TABLE]

It may be shown that Eq. (139) which leads to the magnetic equilibrium (see Eqs. (4-1) in Spitzer (1962))

[TABLE]

where the pressure $p$ corresponds to the overshoot tachocline (at the bottom of the convection zone).

From the Maxwell equation $4\pi j_{\perp}/c=4\pi j_{y}/c=\nabla\times\vec{B}$ we get

[TABLE]

By equating (141) and (142) and integrating in the limits $[B_{tacho},0]$ on the left and $[0,p_{tacho}]$ on the right, we find the repulsive magnetic field induced by the generalized thermomagnetic EN effect:

[TABLE]

where the magnetic field of the generalized thermomagnetic current in the overshoot tachocline “neutralizes” the magnetic field of the core in magnetic white dwarf.

We adopt a model of the stably burning hydrogen shell on the helium core obtained by solving the equations of hydrostatic balance, heat transfer, energy generation and mass conservation (Robinson et al., 1995; Steinfadt et al., 2010). However, the most important feature between the H and He layers for our purposes is not the chemical composition, but the neutralization of the magnetic fields in the core on its boundary.

Taking the density in the tachocline $\rho=\rho_{periph}\leqslant 10^{2}~{}g/cm^{3}$ and $T\sim 10^{6}~{}K$ (see (Bhatia, 2001)), let us estimate the pressure of the nonrelativistic electron plasma (143):

[TABLE]

where the toroidal magnetic field in the tachocline

[TABLE]

By analogy to Eq. (136) for the solar tachocline, we can estimate an important upper bound of the GUP parameter:

[TABLE]

again, assuming experimental precision of 1 part per $10^{n}$, e.g. for $n=2$, where we used $B=B_{tacho}\approx 4.1\cdot 10^{3}~{}T\approx 8\cdot 10^{-12}~{}GeV^{2}$, $L=L_{tacho}\leqslant 10^{2}~{}km\approx 10^{21}~{}GeV^{-1}$, where $L_{tacho}$ is the thickness of the tachocline in the magnetic white dwarf (Kissin and Thompson, 2015). The total bound on $l_{n}\geqslant(\Delta x)_{min}=\alpha_{0}l_{Pl}$ (see Eq. (130)) reads:

[TABLE]

which again indicates an intermediate length scale between the electroweak interactions and the Planck length. The quantum gravity corrections lead to the restraint of the star collapse if the GUP parameter $\alpha=\alpha_{0}l_{Pl}/\hbar$ takes the values between the Planck length and the electroweak interactions scale (see Eqs. (137) and Eq. (147)).

It means that quantum gravity induces the repulsive magnetic fields remarkably well in both the solar (see Eqs. (26)-(27)) and white dwarfs tachoclines (see Eq. (143)). Recalling in addition the EN effect theory for the quantum phase transitions in dyonic BHs (Hartnoll et al., 2007, 2016), one may say that the quantum gravitation and the induced repulsive magnetic field in the generalized thermomagnetic EN effect play an important role in the astrophysical phenomena and the structure of the Universe. This is a separate problem though, and we hope to obtain some results in this direction in the future.

Meanwhile, the part related to the generalized thermomagnetic EN effect by analogy to the solar tachocline near the boundary of the BH is presented below.

\subsubsubsection

Thermomagnetic Ettingshausen-Nernst effect near quantum phase transitions in dyonic black holes

BHs in the centers of galaxies are surrounded by a strong magnetic field due to charged matter around them (see e.g. Aliev and Özdemir (2002); Konoplya (2006, 2007); Konoplya and Fontana (2008); Jamil et al. (2015); Garofalo (2017)). As noted by Johnson et al. (2015), “Understanding these magnetic fields is critical. Nobody has been able to resolve magnetic fields near the event horizon until now”. The authors measured polarization near the event horizon of Sgr A*, the SMBH at the center of our galaxy. The polarization is believed to be a signature of ordered magnetic fields generated in the accretion disk around the BH. The results are expected to explain how BHs accrete gas and launch jets of material into their surroundings.

Such explanation may be related to the known solution of the perturbative Einstein-Maxwell equations depending on the BH mass $M$, magnetic field $B$, and the parameter $K$, which characterizes the above surrounding structure (see Konoplya (2006)). Hence, the presence of the gravitational tidal force from surroundings considerably changes the parameters of the test particle motion (see e.g. Konoplya (2006)): when the magnetic field is weak (strong), it increases (decreases) the radius of circular orbits of particles and the binding energy of massive particles going from a given circular orbit to the innermost stable orbit near the BH. In addition, it increases (decreases) the distance of minimal approach, time delay and bending angle for a ray of light propagating near the BH (see e.g. Gammaldi et al. (2016); Belikov et al. (2016); Lacroix et al. (2016)).

On the other hand, taking into account the DM existence and the possible modifications to general relativistic gravitational wave patterns (see Abbott et al. (2016a, b); Barceló et al. (2017)), let us consider some important thermodynamic features of this BH space-time, which are solutions of the Einstein-Maxwell theory in 3+1 dimensions, and include the quantum gravity effects beyond general relativity.

The matter is the AdS/CFT correspondence (Maldacena, 1999), which demonstrates a close connection between Einstein’s general theory of relativity and quantum physics.

Here we are interested in the system at finite temperature $T$, charge density $\rho$ and with a background magnetic field $B$. The finite temperature is realized in $\text{AdS}_{4}/\text{CFT}_{3}$ by allowing the space-time to contain a BH (Witten, 1998), where the temperature in the field theory is just the Hawking temperature of the BH and the finite temperature dissipation in the field theory is dual to bulk matter fields falling into the BH. The important result of this is the fact that the crosswise flow of heat, associated with the finite temperature dissipation in the field theory, and charge currents in the presence of the magnetic field are obtained by allowing the black hole to carry the thermomagnetic EN effect (see Spitzer (1962, 2006); Rusov et al. (2015)), which in turn produces the large-scale magnetic fields near the BH (see inset in Fig. 26a, and also Eqs. (26)-(27)).

Magnetism, produced by the generalized thermomagnetic EN effect (see the essence of quantum gravity for dyonic BHs by Hartnoll et al. (2007)), plays a fundamental role in the generation of the large-scale magnetic fields of the BH. It pushes the gas away from the BH (see e.g. similarities and differences in the papers by Fukumura et al. (2017) and Contopoulos et al. (2015, 2018)). Therefore, we understand that quantum gravity is not only the base for the GUP and the dyonic BH solution (Hartnoll et al., 2007, 2016), but also gives birth to the repulsive magnetic field induced in the “tachocline” by the generalized thermomagnetic EN effect in compact objects (see e.g. Shapiro and Teukolsky (1983)): our Sun, magnetic white dwarfs, accreting neutron stars (similar to white dwarfs; see the review by Wang (2016)) and dyonic BHs.

A number of intriguing questions appear at this point (see Fig. 26). How do the large-scale magnetic fields of the thermomagnetic EN effect extract the energy from the central BH? Or, to put it differently, how does the ergosphere cause the magnetosphere inside it to rotate, making the outgoing flux of angular momentum result in extraction of energy from the BH? How can the large-scale magnetic fields of the thermomagnetic EN effect be explained with the magnetic connection between the BH and the accretion disk? If there is a magnetic link, then how does the BH exert a torque on the accretion disk, thus transferring energy and angular momentum between the BH and the disk? How do the cycles of accretion from a thin disk and the associated spin-up of the BH alternate with the periods of no accretion and magnetic transfer of energy from the BH to the disk? How is the jet launched as a magnetically dominated outflow with strong magnetic fields accelerating the flow to relativistic velocities?

One can ask a lot of other questions, but first of all, it is necessary to understand and describe in detail the thermomagnetic EN effect – a fundamental phenomenon of crucial importance for powering relativistic outflows of jets from BHs (see Blandford and Znajek (1977); MacDonald and Thorne (1982); Thorne et al. (1986); Punsly and Coroniti (1990)).

In Fig. 26b one can see how the magnetic flux is pushed out of the quickly rotating BH horizon. The nature of this phenomenon is similar to the Meissner effect (see e.g. Bičák et al., 2007; Komissarov and McKinney, 2007; Penna, 2014a, b; Bičák and Hejda, 2015; Gürlebeck and Scholtz, 2017; Hirsch, 2016, 2017), i.e. displacement of magnetic fields in superconductors. Later Bičák and Janiš (1984) (see also Bičák et al., 2007) showed that this was true for all axisymmetric stationary vacuum solutions of the BH electrodynamics. They concluded that their discovery undermines the widely known electromagnetic mechanism of Blandford and Znajek (1977), which, in contrast to the vacuum Meissner effect, is concerned with plasma-filled magnetospheres. Bičák and Janiš (1984); Bičák et al. (2007) believe that this BH Meissner effect could quench jet power at high spins, but we think the actual answer is “yes and no”. Let us show why.

The EN thermomagnetic mechanism, which generates the Meissner effect at Hawking’s BH temperature, is considered one of the most effective ways of extracting the net energy from rotating BHs, for example, a dyonic BH (see Spitzer (1962, 2006); Hartnoll et al. (2007, 2008b, 2008a); Rusov et al. (2015)). Our view of this mechanism is as follows. The spinning BH distorts the poloidal magnetic field $B_{P}$ and induces the poloidal electric field $E_{P}$ and toroidal magnetic field $B_{T}$ (see inset in Fig. 27a), which generate an outward Poynting flux $E_{P}\times B_{T}$ along the magnetic field lines threading the spinning BH. The rotation energy of the spinning dyonic BH is extracted in the form of the Poynting flux. The formal physics of this idea almost completely coincides with the well-known idea of Komissarov (2009) and Beskin (2010), but, surprisingly, it describes not only the BZ-like effect (see e.g. Pan et al. (2017); Contopoulos et al. (2017) and Refs. therein), but also the thermomagnetic EN effect (see Fig. 26b and Fig. 27b). To proceed further, it is natural to ask why doesn’t the relativistic magnetohydrodynamic (MHD) simulation of jets of BHs give any evidence (see e.g. Penna (2014a, b)) that jets are extinguished by the Meissner effect of the BH. In general, why should the Meissner effect exist along with the BZ-like effect, from the point of view of the BH jets existence? Below we demonstrate the possibility of overcoming the contradictions between these effects.

It is important to understand this possibility and the essence of the existence of holographic superconductors. This is due to the fact that in determining whether our model is Type I or Type II superconductors, we are confronted with the limitation that the currents (see the right panel in Fig. 27c) in the model are not a source of electromagnetic fields (see Hartnoll et al. (2008b)). As the material does not produce its own magnetic fields, the applied magnetic field is the actual magnetic field outside the horizon and we can set $B_{T}$. The major benefit of the London equation is the ability to explain the Meissner effect (Meissner and Ochsenfeld, 1933), when the material exponentially pushes out the internal magnetic fields after passing the superconducting threshold – the horizon. However, the most important is the fact that the holographic superconductors (Type II) generate the currents required to expel magnetic fields (the London equation and the associated London magnetic penetration depth) and that the theory can consistently be weakly gauged (see Hartnoll et al. (2008b)).

So, our starting point is the holographic superconductor as a dyonic BH background (see Hartnoll et al. (2008b, a)). This leads to the existence of the thermomagnetic EN effect at low temperatures (the Hawking temperature), otherwise known as the BH Meissner effect, which extracts pure energy near the horizon by the BH spin or by an accretion flow, feeding on dyonic BH jets. Note that this is not the BZ mechanism, but the EN mechanism (see Fig. 27a), which, strangely enough, is one of the most effective ways to extract the rotation energy from spinning BHs. And this argument is reasonable for accreting BH systems, such as AGNs and X-ray binaries, but not in gamma-ray bursts (GRBs) events by the BZ mechanism (see Fig. 27b), which we will discuss below.

If the thermomagnetic EN effect, which gives rise to the Meissner effect at low temperatures (see Fig. 27a), transforms the variable magnetic fields within the holographic superconductor (see Hartnoll et al. (2008b)), it means that if the strength of the external magnetic field changes for some reason (e.g. because of the chaotic variations of the baryon density and, consequently, the ADM density; see Fig. 27c), the variations in baryon density would lead to a rather quick transition from the Meissner effect (Fig. 27a) to the BZ effect (Fig. 27b). Hence, we understand that the BZ effect excludes the external magnetic field of the Meissner effect near the horizon, since, unlike the Meissner effect, it changes the shape of the magnetic field and thereby reveals a mechanism, how the spin energy of a rotating BH may be extracted electromagnetically through the magnetic field that threads the BH horizon (see e.g. Nathanail and Contopoulos (2014); Penna (2014a, b); Contopoulos et al. (2017)). Therefore, the chaotic variations of the flux always penetrate into the thin film (see e.g. Fig. 27b on the right). It is very important that the BZ effect also shows that it is possible to hold this magnetic flux for the duration of the GRB even without accretion onto the BH (see the original Penrose process of Penrose and Floyd (1971), Contopoulos (1984) and the analogous one of Contopoulos et al. (2017)).

So, on the one hand, the duration of a GRB depends closely on the magnetic flux accumulated on the event horizon, and on the other hand, that is actually why we observe a clear exponential decay in the fraction of GRB events (see Contopoulos et al. (2017)). And vice versa, in order to restore the Meissner effect, it is necessary to include the non-chaotic variations of the magnetic field, and thereby to push out the external magnetic field with the help of a current (see e.g. Fig. 27a), the force of which must “neutralize” the strength of the opposite field inside the three-dimensional sample. If neutralization is not complete, then the chaotic variations of the flux always penetrate into the thin film (see e.g. Fig. 27b on the right). This argument is illustrated in Fig. 27c (adopted from Hartnoll et al. (2008b)). As a result, we have a possible transition from the Meissner effect to the BZ effect and back. This means that the appearance or disappearance of the chaotic modulating density of baryons (and correspondingly, the density of DM) on the horizon of a BH is the main basis for the manifestation of the BZ or Meissner effects, respectively.

Ignoring many other questions of the Meissner effect in the BH, we focus on the main result of the theoretical-experimental proof of the fundamental existence of the repulsive magnetic field induced in the “tachocline” by the generalized thermomagnetic EN effect of quantum gravity, which can be evaluated on compact objects: our Sun, magnetic white dwarfs, accreting neutron stars and dyonic BHs.

In this context, let us consider the physics of the so-called holographic principle of quantum gravity. The outstanding paper by Hanada et al. (2014) presents an intriguing argument for the superstring theory as a specific implementation of the holographic principle (Bekenstein, 1973; Hawking, 1973; ’t Hooft, 1993; Susskind, 1995; Maldacena, 1999). Simply speaking, the hologram is a 2D object, but if we look at it under certain light, we can see the image of a 3D object encoded within this 2D surface (see Bekenstein (1981, 1997, 2003, 2007); Brown et al. (2016)). According to these new theories, the whole Universe (or an individual BH, or a white dwarf, or a neutron star) may be a sort of a hologram. It may be thought of as a 3D object with all internal complex systems consisting of “quarks” and “gluons” being described by the laws based on gravity, or as a 2D surface within which the “quarks” and “gluons” are subject to completely different laws like quantum mechanics, and no gravity on the surface.

There is however a problem related to the fact of Hawking radiation from BHs, which would lead to their evaporation and the paradox of information loss (Hawking, 1976). At first it may seem to violate the laws of quantum mechanics. On the other hand, quantum mechanics imposes a strong rule forbidding the information loss – the unitarity principle, which is closely related to other inviolable physical laws such as the energy conservation (see e.g. Banks et al. (1984); Giddings and Shi (2013); Giddings (1992, 2012, 2013); Modak et al. (2015)). And there is a controversy in understanding of the quantum mechanics with and without the unitary processes (see e.g. Chen, Ong and Yeom (2015)). One party suggests that the statement by Banks et al. (1984), according to which the unitarity violation would violate both the locality and the energy-momentum conservation, is a strong argument against non-unitarity of Hawking radiation (Hawking, 1976). Another party relies on the fact that Nikolić (2015) recently showed that the alleged problem with non-unitary Hawking radiation, the problem of violation of either locality or conservation of energy-momentum, does not really exist. This is because the non-unitary time evolution of Hawking radiation is not in contradiction with a generalized concept of unitarity (see Vafa (2014); Nikolić (2015)).

The remarkable paper by Vafa (2014) strongly supports the fact that the unitarity is not a necessary consequence of holography, and that the existence of non-unitary holography can probably be established using the known 2D non-unitary models which should lead to non-unitary AdS3 holographic duals (see Hawking (1976); Vafa (2014)).

We are struck by the fact that non-unitarity is consistent with the idea of holography, wherein quantum gravity is to be constructed in terms of degrees of freedom that are highly nonlocal from the bulk point of view (see Almheiri, Marolf, Polchinski and Sully (2013); Almheiri, Marolf, Polchinski, Stanford and Sully (2013)). A possible unexpected effect is that the properties of non-unitary holography can be experimentally measured precisely on the basis of the fundamental physics of the holographic principle of quantum gravity.

$\bullet$ First of all, we are interested in the properties of the holographic principle of quantum gravity, which make it possible to measure the ratio between the magnetic fields of the 2D surface and the 3D volume of compact objects: our Sun, magnetic white dwarfs, accreting neutron stars and dyonic BHs.

Using the thermomagnetic EN effect (see Spitzer (1962, 2006); Rusov et al. (2015) and Fig. 11), the magnetic pressure of the ideal gas inside e.g. the solar tachocline may be estimated as

[TABLE]

which indirectly shows that, according to the holographic principle of quantum gravity, the repulsive magnetic field of the tachocline exactly “compensates” the magnetic field of the solar core (see Fig. 3) where the projections of the magnetic fields in the tachocline and the core have equal values but opposite directions:

[TABLE]

The pressure of the nonrelativistic electron plasma in the tachocline of cold magnetic white dwarfs may also be estimated:

[TABLE]

Accordingly, the repulsive magnetic field of the tachocline exactly “neutralizes” the magnetic field in white dwarfs (see the analog of Kissin and Thompson (2015)):

[TABLE]

A rough estimate of the pressure of degenerate relativistic electron plasma in the tachoclines of accreting neutron stars and dyonic BHs is also possible:

[TABLE]

The corresponding “neutralizing” magnetic field in accreting neutron stars and dyonic BHs

[TABLE]

where in a fully ionized hydrogen plasma with $Z=1$ the generalized thermomagnetic EN effect leads (see Fig. 27a) to the current density given by

[TABLE]

So our semi-phenomenological result consists in the fact that the repulsive magnetic field of the tachocline, produced by the thermomagnetic EN effect (see e.g. Fig. 27a) on compact objects, is the fundamental consequence of the holographic principle of quantum gravity in the Universe.

$\bullet\bullet$ Second, we are interested in the connection between the thermomagnetic EN effect and the fuzzball effect near the BH boundary.

As we know, according to the AdS/CFT correspondence (Maldacena, 1999) or, generally, the gauge/gravity correspondence (Horowitz and Polchinski, 2006), the quantum gravitation theory in the volume is mathematically equivalent to the quantum field theory on its border. Therefore if the thermomagnetic EN effect on the border of the BH is physically equivalent to the Meissner effect (see Fig. 27a), the EN effect under low Hawking temperatures (or the unitary Meissner effect) is a consequence of the so-called “energetic curtain” (Braunstein et al., 2009, 2013) or “fuzzball” (Almheiri, Marolf, Polchinski and Sully, 2013; Almheiri, Marolf, Polchinski, Stanford and Sully, 2013; Marolf and Polchinski, 2013; ’t Hooft, 2017a) formation upon the so-called fuzzball (see Mathur (2005); Almheiri, Marolf, Polchinski, Stanford and Sully (2013); ’t Hooft (2017b); Barceló et al. (2017); Chakraborty and Lochan (2017) and Refs. therein), which blocks the non-unitary Hawking radiation by the BZ-like effect (see Fig. 27b). This is because the non-unitary time evolution of Hawking radiation (see Hawking (1976); Vafa (2014); Nikolić (2015)) does not contradict the generalized unitarity principle (see Hartle (1998); Nikolić (2009, 2012, 2014, 2015)), and thus does not contradict the unitary time evolution of the holographic Meissner effect (see Fig. 27).

Note that in contrast to the firewall (Almheiri, Marolf, Polchinski and Sully, 2013; Almheiri, Marolf, Polchinski, Stanford and Sully, 2013; Marolf and Polchinski, 2013; ’t Hooft, 2017a) argument with the help of the so-called ‘validity of effective field theory’ (Almheiri, Marolf, Polchinski and Sully, 2013; Almheiri, Marolf, Polchinski, Stanford and Sully, 2013; Marolf and Polchinski, 2013), an infalling shell-object of fuzzball never falls into the trap of its own horizon, and thus avoids any problem with causality, as opposed to the firewall (Mathur, 2009c, 2017a, 2017b).

The surprising result is that it excludes the information loss due to non-unitarity completely, thus undermining the very logic used for the paradox formulation (see Braunstein et al. (2013)). One should also keep in mind that our indirect experimental results of the thermomagnetic EN effect in tachoclines of the Sun (see Eq. (149)) and white dwarfs (see Eq. (151)) may become a direct proof if, for example, the new research succeeds in getting the information on the structure of the BH fuzzball from the observations of gravitational waves (see e.g. Abedi et al. (2017); Zhang (2017) and Refs. therein). According to our holographic model of the fuzzball, it is possible to discover the gravitational echo-effects as a result of the macroscopic quantum gravity effects in BHs beyond general relativity (see Barceló et al. (2017) and Refs. therein).

$\bullet\bullet\bullet$ Third, we are interested in the connection between the holographic Meissner-like effect near the BH boundary, which produces the “fuzzball” phenomenon, and the Higgs phenomenon in particle physics.

We start with understanding that the holographic Meissner-like effect near the BH boundary – fuzzball surface (see Fig. 27a), which is essentially identical to the EN effect at Hawking’s BH temperature, is a consequence of quantum gravity by the gauge/gravity correspondence. It is then easy to show that the EN effect is physically identical to the Meissner-like effect not only for low temperatures, but for any (gradients of) temperature. It exactly compensates the magnetic fields between the 2D surface of the “tachocline” and the 3D volume of the core of compact objects. Since the exact neutralization of the 2D and 3D fields (see Fig. 27c) is determined by one of the properties of the holographic principle of quantum gravity (spontaneous breaking of electroweak symmetry), it leads to a surprising result: the EN effect is absolutely identical to the Higgs phenomenon in particle physics (see Wilczek (2000, 2005)).

$\bullet~{}\bullet~{}\bullet~{}\bullet$ Forth, we are interested in the connection between the Higgs phenomenon in particle physics (identical to the thermomagnetic EN effect) and the quantum phenomenon of DM particles.

The Higgs discovery showed the scientists a new top-priority direction in solving one of the major puzzles in cosmology – the nature of DM. The experimental results pushed physicists away from the Z-bozon, so the only particle able to interact with DM directly was the Higgs particle (see Lykken and Spiropulu (2013)). Taking into account the cogitative history of the Higgs mechanism (see e.g. Lykken and Spiropulu (2013); Ivanov (2017) and Refs. therein), the Higgs boson (see Fig. 28a) is naturally a hot topic of new physics beyond the Standard Model through the Higgs window, with the strongest indications coming from its capacity to accommodate DM and provide a viable explanation of the baryon asymmetry of the Universe.

Higgs mechanism problem. In our view, the theory of electroweak interactions (Weinberg, 1967; Salam and Ward, 1964; Glashow, 1961) and the related Higgs mechanism (Higgs, 1964b, a) contain a number of problems, revealed, among other things, by some inconsistencies with the recent experimental results (see Chatrchyan et al. (2012); The LHC Higgs Cross Section Working Group (2013); Aad et al. (2016)). In particular, the cited papers mention the experimental observations of the Higgs boson decay channels gathered in the report by Particle Data Group (Patrignani and Particle Data Group, 2016). As it is seen from these results, the total weak isospin of the particles in the final state of the decay may be integer only. At the same time, the Higgs field in the Standard Model (Weinberg, 1967; Patrignani and Particle Data Group, 2016) is transformed according to the fundamental representations of the SU(2) group, and is two-component. So, according to the Standard Model, the Higgs boson has the weak isospin of 1/2 (see e.g. Peskin and Schroeder (1995); Ryder (1996)). The analysis of the $\beta$-decay as well as various channels of lepton decays (Patrignani and Particle Data Group, 2016) leads to a conclusion that the weak interaction conserves the weak isospin. Since the carriers of other interactions have zero weak isospin, it must conserve in the processes related to other interactions as well.

On the theory side, we believe that the problem of the Standard Model is the introduction of the “new” non-gauge interactions. Particularly, the nonzero vacuum mean of the Higgs field is achieved due to the non-gauge $\varphi^{4}$ interaction. The only manifestation of such interaction being discussed is the spontaneous breaking of symmetry (Goldstone et al., 1962). So it is unclear how the existence of such interaction may be confirmed experimentally. The same is applicable to the Yukawa interaction of fermion fields with the Higgs field (Weinberg, 1967; Peskin and Schroeder, 1995; Ryder, 1996), which provides them with mass. This is not a result of some symmetry localization, and thus cannot be reduced to one of the known interactions!

Another point is the “unnatural” sign of the Lagrangian term quadratic in Higgs field. Although it does not lead to nonphysical results, the question remains of why there is only one field with such properties. And the only argument in favour of such notation seems to be the appearance of mass for the gauge particles (see e.g. Grojean (2007); Quigg (2009); Troitsky (2012); Grojean (2014)).

While considering the multiparticle fields (see Volkotrub et al. (2015); Chudak et al. (2016); Merkotan et al. (2017)), it was noticed that the dynamic equation of the two-particle gauge field looks similar to the dynamic equation of the $\varphi^{3}$-theory. The difference is that instead of the squared mass there is an operator which may have non-negative eigenvalues under certain conditions. The two-particle gauge field operators describe the creation and annihilation of particles, which are the bound states of the gauge bosons. Therefore, the self-action of the two-particle gauge field is the manifestation of the interaction between quanta of the non-Abelian gauge field and does not require the addition of an extra interaction. So in the model of multiparticle fields we have both components necessary for the spontaneous violation of symmetry. The Higgs boson is then (like in some other models) not an elementary particle, but a bound state. This possibility was also pointed out by Peter Higgs (Higgs, 1964b, a). However, he expected the scalar field, which violates the symmetry, to consist of fermion fields, and not of gauge bosons. Hoh (2016a, b) considers the Higgs boson as a bound state of gauge bosons, which are in the state of confinement. In the papers by Volkotrub et al. (2015); Chudak et al. (2016) the two-particle gauge field describes the gluon and quark confinement. The field with spontaneous breaking of symmetry describes the creation and annihilation of the bound states of gauge bosons without confinement, i.e. the bound states with a finite binding energy. Still, although such fields have nonzero vacuum mean, they cannot lead to the appearance of mass for the gauge bosons, since the scalar representation of the internal symmetry group is realized upon them. As a result, they cannot interact with the single-particle gauge field and produce its mass.

For this reason in the present paper the aim is to consider the two-particle gauge field with the vector representation of the internal symmetry group. Since we are interested in the Higgs mechanism, we choose the SU(2) group (Merkotan et al., 2017).

Multiparticle fields and the Higgs mechanism – a brief discussion of the main results (see Merkotan et al. (2017)).

In contrast to the (incomplete) Standard Model, the Higgs boson here is supposed to have integer weak isospin, which is consistent with the experimental data (Patrignani and Particle Data Group, 2016) on its decay channels. It is considered as a bound state of W+ and W- bosons – the particles with weak isospin of 1. Such a bound state may possess a weak isospin of 0 or 1 or 2, depending on three terms in expansion into irreducible tensors (see Eq. (7) in Merkotan et al. (2017)). The channel of decay into two photons (see Patrignani and Particle Data Group (2016)) means that the Higgs boson may be in the state with weak isospin of 0. The channels of decay into two particles with 1/2 weak isospin, e.g. into electron and positron, add the possibility to observe the Higgs boson in the state with weak isospin of 1. The four lepton channels add the possible value of 2.

So, the known decay channels indicate that the Higgs boson is not an eigenstate of the weak isospin, and its measurement may yield only the integer values of 0, 1 or 2. This property is represented by Eq. (7) in Merkotan et al. (2017). It is essential that this model considers the self-action of the Higgs field (which provides the nonzero vacuum value) not as an independent non-gauge interaction, but as a manifestation of the non-Abelian gauge field. Namely, since the Higgs boson is considered to be a bound state of W+ and W- bosons, the interaction of Higgs bosons is a consequence of the non-Abelian weak SU(2) interaction of gauge bosons. This is formally seen from the derivation of equations for the two-particle gauge field in Sect. 3 in Merkotan et al. (2017). The authors acknowledge that this procedure is somewhat artificial, but it lets one describe some important experimental details. In particular, the use of the similar procedure in the papers by Volkotrub et al. (2015); Chudak et al. (2016) made it possible to describe the quark and gluon confinement. So it lets one derive the Lagrangian term with “unnatural” sign of the quadratic mass, instead of simply introducing it, as it is done in the common Standard Model.

Unfortunately, the suggested model does not solve the problem of non-gauge introduction of the Yukawa interaction between fermion fields and the Higgs field into the Standard Model. From the physical point of view, fermions interact with W bosons, and the Higgs boson consists of the W bosons in this framework, so such interaction should indeed take place. It is unclear how to introduce it on the basis of the gauge principle though. All fundamental fermion fields interact with W bosons, so they can interact with each other through the Higgs field. On the one hand, it explains the neutrino mixing, which leads to neutrino oscillations (see Fukuda, Y. et al. (1998) (Super-Kamiokande Collaboration); Ahmad, Q. R. et al. (2001) (SNO Collaboration); Ahmad, Q. R. et al. (2002) (SNO Collaboration)). On the other hand, it is unclear why such oscillations exist for neutrinos only, i.e. why there are no oscillations between electron and muon, both of which may interact with the Higgs field through W bosons. This field should be “mixing” them just like the corresponding neutrinos.

Another problem not solved in the present paper is that it is impossible to divide the gauge fields into the charged field of W bosons and the uncharged field of Z boson in the gauge-invariant way. So in order to move to a new gauge, it is necessary to switch to the real fields $A_{a_{1},g_{1}}(x)$ (see Sect. 2 in Merkotan et al. (2017)), do the gauge transformations and then separate the fields of W and Z bosons. This is the view of the W+, W- and Z fields in the present paper.

According to Eq. (7) in Merkotan et al. (2017), in addition to the discussed antisymmetric and symmetric traceless irreducible terms, the two-particle gauge field contains a scalar part. According to Eqs. (15)-(42) in Merkotan et al. (2017), this part also has nonzero vacuum value, which appears as a result of spontaneous breaking of symmetry. So it is natural to call such a field the Higgs field, scalar with respect to interior indices. However, this field cannot interact with the gauge field and provide the mass for its components. The particles of this field carry no electric charge, because the field is real. They have no interior SU(2) indices, so they do not participate in weak interactions. Since they are considered as the bound states of gauge SU(2) bosons, such particles cannot participate in strong interactions. At the same time, such a scalar field contributes into the gravitational field through the energy-momentum tensor. Thus, the Higgs bosons, scalar with respect to interior indices, may be considered a candidate for a part of DM!

$\bullet\bullet\bullet\bullet\bullet$ And finally, fifth, we are interested in the link between the physics of the emission of gravitational waves and the generalized thermomagnetic EN effect of ADM around a BH.

In this case we are looking for the possible signs of the quantum gravity alternatives near the BHs horizons through the gravitational waves emission from the BH mergers observed by the advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO) (Abbott et al., 2016a, b; Giddings, 2016). Interestingly, by creating a phenomenological pattern for the consecutive echoes from exotic quantum mirrors, expected in the firewall or fuzzball paradigms, Abedi et al. (2017) obtained the first preliminary evidence of such echo signals in aLIGO data on BH mergers (false detection probability was 0.011, and the significance level was 2.5$\sigma$).

In our view, these quantum reflected echo signals (see Fig. 1 in Abedi et al. (2017)) are possible not only from the BH mergers, but also from the single SMBHs, where the ADM (and accordingly, baryons) modulation determines the formation and existence of the galactic gravitational waves (see Eq. (2.1) of Allen and Romano (1999); Eq. (2) of Abbott et al. (2016b); Eqs. (4.1) and (4.3) of Raidal et al. (2017)).

But here it is very important to understand that the modulations of ADM (and, accordingly, baryons) predetermine not only the emergence of galactic gravitational waves, but also the modulation of gravitational waves. If the changes in the gravitational field are non-chaotic, then the EN fuzzball is formed near the BH boundary (see Figs. 29a, 27a, 33b). Let us note that the EN, Meissner and Higgs effects are absolutely identical here! On the other hand, if the changes in the gravitational field are chaotic, then the surface of the BZ fuzzball is determined by the BZ-like effect (see Fig. 27b), the EN fuzzball disappears instantly, in the absence of which the BZ fuzzball becomes “visible”, e.g. through the gamma ray burst (GRB) from the Higgs decay (see Figs. 29b, 27b, 33c).

Since the holographic principle of quantum gravity, consisting in the relation between the magnetic fields of the 2D surface of “tachocline” and 3D volume of compact objects, is the consequence of the generalized thermomagnetic EN effect on BHs, the physics of the gravitational waves modulation is determined by the physics of the EN or BZ fuzzball surface near the BH boundary. Simply put, this means that if these magnetic fields are exactly “holographically” equal (see Fig. 27c), the “mirror” of the EN fuzzball appears near the BH boundary with a completely extruded magnetic field from the EN fuzzball surface (see Fig. 27a and Fig. 29a). If the holographic magnetic fields do not compensate each other, the BZ fuzzball surface appears as a “mirror with holes” instead (see Figs. 29c, 33b), and it cannot push the magnetic field out of the BH (see Figs. 27b and 29b). This simple and clear model assumes that the “mirror” in the form of the EN fuzzball surface predetermines the appearance of Hawking radiation and signals reflecting the quantum echo signal (see Fig. 29a), while the BZ fuzzball surface near the BH boundary causes “evaporation” gravitational waves (see Fig. 29b). This is not so far-fetched, because the “evaporation” of the gravitational waves that appear during the transition from the EN to BZ fuzzball surface is predetermined by taking into account any complementarity of the BZ fuzzball through their collective dynamics (see Figs. 29b, 33b), in which the problem of the information paradox, known as the obvious contradiction between the general theory of relativity and quantum mechanics, disappears completely!

ADM modulation and magnetic fields near the BH boundary.

Taking into account the properties of dyonic BHs (Hartnoll et al., 2007), we find the repulsive magnetic field induced by the generalized thermomagnetic EN effect, where the wind from the BH accretion disk is governed by magnetic pressure, which within the wind must be as follows (see analogous equations by Shakura and Sunyaev (1973); Proga (2003); Miller et al. (2016)):

[TABLE]

where $B_{tacho}$ is the magnetic field of baryon matter, $T_{tacho}$ is the absolute temperature and $n_{tacho}=\rho/m$ is the baryon number density.

This means that the existence of the repulsive magnetic field induced in the “tachocline” near the BH boundary (as well as on the other compact objects, see Figs. 26b, 30) plays an essential role in the acceleration and collimation of the wind, as well as the accretion disk dynamics and evolution (see e.g. Miller, 2017). Let us explain “where the disk magnetic field comes from”.

The accretion disks have to transfer the angular momentum in order for the matter to move radially towards a compact object. The inner viscosity of the magnetic processes (see Shakura and Sunyaev, 1973; Balbus and Hawley, 1991, 1998; Hawley et al., 1995; Pariev and Colgate, 2007; Pariev et al., 2007; Colgate et al., 2014, 2015), and disk winds (see Blandford and Payne, 1982; Miller, Raymond, Fabian, Steeghs, Homan, Reynolds, van der Klis and Wijnands, 2006; Miller et al., 2008; Miller, 2017; Fukumura et al., 2017, and Refs. therein) may in principle transfer the angular moment, but we still lack for the proof that it happens.

It becomes clearer with our alternative (see Figs. 26b, 30), which is substantially different from the other candidate processes, e.g. Blandford and Payne (1982), where the gas might escape along the toroidal magnetic field lines, transferring the angular momentum and allowing the mass transfer through the disk and then accreting to the BH (see differences between Fig. 1 of Pariev and Colgate (2007); Pariev et al. (2007) and Fig. 26b). In the generally accepted Shakura-Sunyaev disk model (Shakura and Sunyaev, 1973), depending on the initial conditions, the method of matter compression and the degree of ordering of the magnetic field, the energy of matter is $\varepsilon=3\rho(kT/m)+bT^{4}=\rho(v_{s}^{2}/2)$, as well as can sufficiently exceed it, reaching the magnetic pressure value $(B^{2}/8\pi)\sim\rho v_{s}^{2}/2=\rho(GM/R)$ ($energy/cm^{3}$). In this last case, the stress ($w_{r\varphi}=|B_{r}\times B_{\varphi}/8\pi|$) and the efficiency of the angular momentum transport are so high that the radial accretion will occur (see Shakura and Sunyaev (1973) and Refs. therein).

The essence of fundamental magnetic processes associated with quantum gravity and the generalized thermomagnetic EN effect in the “tachocline” near the BH boundary may be described rather simply as follows. According to our understanding, one of the fundamental effects of the holographic principle of quantum gravity is the existence of “tachoclines” in all stellar objects in the Universe, including all galaxies and, of course, our Sun, magnetic white dwarfs, neutron stars and BHs of the Milky Way (see Sect. 3).

Let us recall that the solar radiation zone rotates approximately like a solid, and the convection zone has a differential rotation. This leads to the formation of a very strong shear layer between these two zones, called the tachocline. Similar physics of the tachocline exists in a BH. As a result, the tachocline shear layers produce virtually empty MFTs (see Sect. 3.1.2), anchored in the BH tachocline and rising to the surface of the disk (see Fig. 30b,d). Since it is known that disk parts rotate around a BH at different speeds that reinforce the fields (Fig. 30a), this turns the accretion disk into a vortex, pulling the substance into a BH and fueling winds that blow some of it outwards (see Fig. 30b).

So, if the physics of winds in the disk is identical to the physics of practically empty MFTs, which start from the tachocline and rise to the outer (Fig. 30c,d) or inner (Fig. 30a,b) part of the disk, then it means that the generalized thermomagnetic EN effect produces the magnetic tubes or the so-called winds and, on the one hand, the substance current flowing in the direction of the pole of the poloidal (meridional) field of the tachocline in the form of jets.

If we recall that the variations of the toroidal magnetic field and ADM density in the BH tachocline exactly anticorrelate with each other (see Eq. (159)), this means that in strong fields the magnetic tubes rise only inside the disk, which is equivalent to the disappearance of winds, and thus they slow down the speed of the accretion. And the opposite, in weak fields the magnetic tubes rise out of the disk surface (see Fig. 30c), which is equivalent to observing the disk winds, and thus they increase the speed of the accretion.

Here a remarkable connection between the density $\rho_{accr}$ of the BH accretion material and the toroidal magnetic field $B_{tacho}$ of the tachocline appears:

[TABLE]

where $j_{\perp}$ is the density of the poloidal (meridional) current towards the BH pole, which is perpendicular to the toroidal magnetic field $B_{tacho}$ in the tachocline due to the thermomagnetic EN effect (see Eq. (20), inset in Fig. 27a, and also Fig. 6a).

When the magnetic pressure is large at low densities and high temperatures, the low densities of the BH accretion material and high magnetic fields in the tachocline lead to the formation of highly collimated spectrally-hard jets, but without a “visible” magnetic tube (jet in AGN in Fig. 30c). And vice versa, when the magnetic pressure is relatively weak at high densities and low temperatures, the high density of the BH accretion material and relatively low magnetic fields in the tachocline lead to the formation of strongly collimated spectrally-soft jets and a visible magnetic tube, i.e. a visible disk “wind” (jet in BHB in Fig. 30c,d).

So, the resulting conclusion looks quite clear: the physical nature of the generalized thermomagnetic EN effect is the cause of both the formation of magnetic tubes rising from the BH tachocline to the disk and the formation of meridional currents in the direction of the BH pole, which generate jets, for example, in AGN or BHB.

Since the physics of magnetic tubes, rising from the BH “tachocline” to the disk, is almost identical, in our view, to the physics of visible or “invisible” disk winds, below we consider the common properties of the generalized thermomagnetic EN effect and the known models of disk winds. Curiously enough, the latter are associated with the mechanism of DM variations around a BH.

Such winds may arise from various processes, which makes their sources disputable (see e.g. Fukumura et al. (2017) and Refs. therein). However, the X-ray spectroscopic data and analysis of the wind associated with the X-ray binary (XRB) GRO J1655-40 (Miller, Raymond, Fabian, Steeghs, Homan, Reynolds, van der Klis and Wijnands, 2006; Miller, Raymond, Homan, Fabian, Steeghs, Wijnands, Rupen, Charles, van der Klis and Lewin, 2006; Miller et al., 2008, 2012; Kallman et al., 2009; Luketic et al., 2010) argued in favour of the magnetic origin, excluding all candidate processes except for the following two: the semi-analytic MHD wind model of Fukumura et al. (2017) and the MHD outflow model of Chakravorty et al. (2016). Plus our model of the generalized thermomagnetic EN effect.

The observations indicate that disk winds and jets in X-ray binaries are anticorrelated (Miller, Raymond, Homan, Fabian, Steeghs, Wijnands, Rupen, Charles, van der Klis and Lewin, 2006; Miller et al., 2008, 2012; Neilsen and Lee, 2009; King et al., 2012, 2013; Ponti et al., 2012). This indicates a link between disk properties, magnetic field configurations and outflow modes, the repulsive magnetic field induced by the generalized thermomagnetic EN effect (see Eq. (155)), which produces the variations of the magnetic fields near the BH boundary (see e.g. Eatough et al. (2013); Zamaninasab et al. (2014); Johnson et al. (2015)),

[TABLE]

as well as the variations of the ADM density $n_{tacho}^{ADM}$ and baryon matter $n_{tacho}$,

[TABLE]

determined by the modulations of the ADM density $n_{tacho}^{ADM}$.

For these magnetic field configurations, let us point out some important features of the density variations of the dark matter $n_{tacho}^{ADM}$, that can distinguish between “cold” and “warm” wind solutions from near-Keplerian accretion disks.

It is known (see Chakravorty et al. (2016)) that the X-ray spectra of BH X-ray binaries (BHBs) contain blueshifted absorption lines, which means the presence of outflowing winds. The observations also show that the disk winds are equatorial and they mostly occur in the Softer (disk dominated) states of the outburst, being less in the Harder (power-law dominated) states.

The properties of the accretion disk are used to infer the driving mechanism of the winds (see e.g. Neilsen and Homan (2012) and Refs. therein). And more or less prominent winds through the various states of the BHB have been interpreted as a variation in the magnetic driving mechanism of the wind (Miller, Raymond, Fabian, Steeghs, Homan, Reynolds, van der Klis and Wijnands, 2006; Miller, Raymond, Homan, Fabian, Steeghs, Wijnands, Rupen, Charles, van der Klis and Lewin, 2006; Kallman et al., 2009; Neilsen and Homan, 2012).

In our case the ADM density $n_{tacho}^{ADM}$ modulations, associated with the generalized thermomagnetic EN effect (see Eqs. (158), (159)), lead to the variations of the toroidal magnetic fields in the accretion disk (see Fig. 30). In order to understand the main motivation for the toroidal magnetic field variations in the accretion disk, forming the “cold” and “warm” wind, it is necessary to discuss the difference between the winds and jets from accretion disks.

Given Eq. (158) and the high (low) magnetic pressure, the density of ADM is relatively low (high), and the gas temperature is high (low). This means that with the high density of ADM and low temperature, the magnetic driving mechanism produces the accretion disk wind, which is equatorial (see Fig. 26c, and the analogous physics of the X-rays of axion origin in Figs. 9a, 6a, and 30b,d) and occurs in the soft states of the BHB outbursts (see e.g. the analogous model by Chakravorty et al. (2016)). Alternatively, with the low density of ADM and high temperature, the magnetic pressure is rather high and, consequently, the magnetic driving mechanism produces the weak or less prominent wind in the hard states (AGN jets).

Either way, the nonrelativistic disk winds and relativistic jets are anticorrelated, since the relativistic AGN jet, induced by the vertical toroidal magnetic field (see Fig. 30) and collisions between ADM and nuclei in the close vicinity of a SMBH (see e.g. the analogous model by Lacroix et al. (2016)), is determined by the modulations of the ADM density (see Eq. (158)):

[TABLE]

This eventually means that the strong magnetic fields near the BH boundary are caused by quantum gravity of the dyonic BH (Hartnoll et al., 2007), which determines the existence of the generalized thermomagnetic EN effect (see Fig. 27b and Spitzer (1962, 2006); Hartnoll et al. (2007); Rusov et al. (2015)). So we understand that the major effect of quantum gravity here is that the initial acceleration (deceleration) of the disk winds and BHB flares (AGN jets) originate from less (more) strong magnetic fields in the accretion disk near the BH boundary (see also Eq. (159), Fig. 30 and inset in Fig. 27a), which predetermine the modulations of the ADM density – the process connected with the variability of the accretion flows, nonrelativistic disk winds and relativistic BHB or AGN jets.

4.1.2 Who generates and controls the modulation of dark matter in the supermassive black hole?

The basic idea is that the modulation of the luminous AGN, associated with the periodical variability, is supposed to be the modulated accretion of baryon matter onto the SMBH, which modulates ADM of baryon nature in the SMBH. Here are some points in support of such a scenario through Occam’s razor:

$\bullet$ It is known that “collisions and interactions between gas-rich galaxies are thought to be pivotal stages in their formation and evolution, causing the rapid production of new stars, and possibly serving as a mechanism for fueling supermassive black holes” (Goulding et al. (2018); see also Di Matteo et al. (2005); Gatti et al. (2015); Koss et al. (2018); Steinborn et al. (2018); Storchi-Bergmann and Schnorr-Müller (2019)).

$\bullet$ A significant result of the last decade is the discovery that the “…the mass of the central black holes (BHs) and properties of the host galaxies, notably the stellar bulge mass or central stellar velocity dispersion”(Bogdán and Goulding, 2015), as well as the total gravitational mass of the host galaxy, or the mass of the ADM halo, closely correlate with each other (see e.g. Magorrian et al. (1998); Gebhardt et al. (2000); Ferrarese and Merritt (2000); Tremaine et al. (2002); Gültekin et al. (2009); McConnell and Ma (2013); Bogdán and Goulding (2015); Zahid et al. (2018); Argüelles et al. (2019)).

$\bullet$ The idea of the galaxies spending most of their life in star formation and in the main sequence (see Elbaz et al. (2011); Stanley et al. (2017)) also suggests that the star formation and BH accretion are closely related (see Alexander and Hickox (2012); Chen et al. (2013); Hickox et al. (2014); Asmus et al. (2014); Thacker et al. (2014); Stanley et al. (2017); Harrison (2017); Cresci and Maiolino (2018); Yang et al. (2019)).

$\bullet$ Another possible evidence of the causal link between the rising of the AGN activity and the quenching of a galaxy can be obtained from the observed correlation of the star formation rate with the AGN luminosity (see Shao et al. (2010); Lutz et al. (2010); Bonfield et al. (2011); Harrison et al. (2012); Hickox et al. (2014); Gürkan et al. (2015); Bernhard et al. (2016); Stanley et al. (2018); Dai et al. (2018); Brown et al. (2019); Aird et al. (2019)).

$\bullet$ The most important point is the fact that “understanding the relationship between the galaxies in which the active galactic nuclei (AGNs) are located and the halo of dark matter in which they are located is the key to limiting how black hole refueling is triggered and regulated”(Leauthaud et al., 2015; Netzer, 2013; Gatti et al., 2016; Ballantyne, 2017; Padovani et al., 2017; Georgakakis et al., 2018; Outmezguine et al., 2018).

These points are connected with the physics of the baryon ADM modulation in the SMBH. In order to solve this problem, it should be remembered that the question of how the AGN feedback controls the star formation and the growth of luminous galaxies (see Asmus et al. (2015)) is solved in a rather straightforward way for the modulation of the luminous AGN (see e.g. McNamara and Nulsen (2007); Fanidakis et al. (2013)). The heating and cooling of gas in galactic clusters coupled by AGN feedback, which suppresses the star formation and luminous galaxies growth, necessarily using the AGN clustering (see Fanidakis et al. (2013); Retana-Montenegro and Röttgering (2017)), may be interpreted in terms of their distribution in host ADM haloes (see Fanidakis et al. (2013); Hickox et al. (2016)).

A rather natural question arises here: if the modulations of the ADM density at the GC give rise to the modulations of the ADM halo density near the solar surface – through the density waves (see e.g. Lin and Shu (1964); Toomre (1977); Bertin and Lin (1996); Dobbs and Baba (2014); Shu (2016)), crossing the spiral arms (see Binney and Tremaine (2008); Baba et al. (2013); Sellwood and Carlberg (2014)), is there any “experimental” link between the ADM density modulation in the solar interior and the sunspot number?

Until now, “an implicit assumption in modeling the tracer density profile was that the local neighborhood is axisymmetric, and long-lived spiral patterns in stellar disks are in dynamic equilibrium” (Buch et al., 2019). However, according to Buch et al. (2019), the growing evidence in Gaia Collaboration (2016, 2018) data for asymmetry in the vertical number counts (Widrow et al., 2012; Bennett and Bovy, 2018), vertical waves in the disk at the Sun position (Purcell et al., 2011; Gómez et al., 2013; Carlin et al., 2013; Widrow et al., 2012, 2014; Xu et al., 2015; Gómez et al., 2017; Carrillo et al., 2018, 2019; Laporte, Gómez, Besla, Johnston and Garavito-Camargo, 2018; Laporte, Johnston, Gómez, Garavito-Camargo and Besla, 2018) and the kinematic substructure (Antoja et al., 2018; Myeong et al., 2018; Necib et al., 2018) warrants a closer look at sources of disequilibrium in the solar neighborhood. Additionally, it is necessary to include the indicators that connect the movements of tracers in different directions, for example, vertical density waves propagating through the spiral arms of our galaxy (see e.g. inset in Fig. 31).

From here we understand that the ADM halo density variability is a consequence of the existence of sources of disequilibrium in the solar neighborhood based on the vertical density waves in the Galactic disk (see Purcell et al. (2011); Gómez et al. (2013); Carlin et al. (2013); Widrow et al. (2014); Carrillo et al. (2018, 2019); Laporte, Gómez, Besla, Johnston and Garavito-Camargo (2018); Laporte, Johnston, Gómez, Garavito-Camargo and Besla (2018)). The passing satellite galaxy or a ADM sub-halo excites vertical, radial and azimuthal coherent oscillations of the stellar disk of the Milky Way from the solar neighborhood towards the Galactic anti-center, where the vertical wave perturbations act in the direction perpendicular to the Galactic plane (see inset in Fig. 31; see also Fig.4 in Purcell et al. (2011)).

This asymmetry as a source of disequilibrium becomes more evident when compared with the well-known model of the Milky Way disk response to the tidal interaction with the Sagittarius dwarf spheroidal galaxy (Sgr; see Purcell et al. (2011)), that leads to the galactic north-south asymmetry (in both the number and the mean vertical velocity distribution of stars (Widrow et al., 2012)) and the vertical wave-like behavior of modes (see e.g. Fig. 4 in Purcell et al. (2011)) penetrating in the Milky Way disk. Not surprisingly, it is qualitatively consistent with the observations (see Purcell et al. (2011); Widrow et al. (2012); Gómez et al. (2013)).

Note that the vertical perturbations in the galactic disk are the result of the orbiting Sgr satellite $\rightarrow$ host DM halo $\rightarrow$ disk interaction. The reason for this has been explained in great detail by Vesperini and Weinberg (2000), and also by Gómez et al. (2016). The main driving force behind this perturbation is low-mass, low-velocity fly-by encounter, which intersects the plane of the disk at a relatively large galactocentric distance. Although the Sgr satellite is not massive enough to directly perturb the galaxy disk, the density field of the host DM halo responds to this interaction, and consequently, the asymmetric features of the density perturbations are transmitted to the internal parts of the primary system, acting on a deeply submerged galactic disk. Such vertical perturbations as a natural consequence of dynamic friction (Chandrasekhar, 1943; Weinberg, 1985, 1986), which excites the DM inside the host halo, can cause the formation of vertical structures of the galactic disk (Weinberg, 1995, 1998; Vesperini and Weinberg, 2000).

Considering the latest $\sim$2 Gyr evolution of the Milky Way, according to Laporte, Gómez, Besla, Johnston and Garavito-Camargo (2018); Laporte, Johnston, Gómez, Garavito-Camargo and Besla (2018), it may be particularly affected not only by the Sgr satellite, but also by the Large Magellanic Cloud (LMC) at the same time. Skipping the complex but remarkable work of Laporte, Gómez, Besla, Johnston and Garavito-Camargo (2018); Laporte, Johnston, Gómez, Garavito-Camargo and Besla (2018), one can simply justify that the physical cause is the same as in the works by Weinberg (1995, 1998); Vesperini and Weinberg (2000): the addition of force from the DM halo induced by the resonant interaction between the DM halo of the Milky Way and both the Sgr satellite and the Magellanic clouds (see e.g. Weinberg (1998)) may be sufficient to explain the observed warp (see Weinberg and Blitz (2006)), which causes the observed vertical density perturbation in the direction of the Milky Way disk (see analogous inset in Fig. 31). In fact, Laporte, Gómez, Besla, Johnston and Garavito-Camargo (2018); Laporte, Johnston, Gómez, Garavito-Camargo and Besla (2018); Laporte et al. (2019) showed that the nonlinear combination of the effects caused by LMC and Sgr remains dominant due to the Sgr effect, which is the most convincing initiator and the main architect of the spiral structure and vertical perturbation of the Milky Way stellar disk.

As will be described in detail, in addition to the large timescale influence of the Sgr satellite and LMC towards the center of the Milky Way, we are interested in the small timescale interaction of the orbital satellite of AGNs $\rightarrow$ bar/host DM halo $\rightarrow$ accretion disk.

Let us start with the condition of birth and variability of AGNs. On the one hand, it is believed that radiation from the AGN is the result of matter accretion onto a SMBH (see Shakura and Sunyaev (1973); Rees (1984)) in the center of its host galaxy. On the other hand, it is known that the luminosities of quasars and other AGNs vary from X-rays to radio wavelengths and at time scales from several hours to many years. It is clear that the matter of a quasar emits light not by command, but by virtue of the processes occurring in it. The fact of synchronism in all points of the region, that is simultaneity of change in conditions and magnitude of the radiation, indicates the compactness of this quasi-star object. It should be also kept in mind that the light from distant quasars comes to us very “reddened”. Funny enough, the wavelength increased due to the redshift, as if deliberately to pass through the Earth atmosphere and be registered in the instruments. In this regard we are interested in the evolution of the ultraviolet luminosity function of AGNs between redshifts z = 4 and 7 using the X-ray/near infrared selection criterion (see Kolodzig et al. (2013); Giallongo et al. (2015); Kulkarni et al. (2019)), especially in the evolution of the luminosity function of the Magellan quasars (see e.g. Kozłowski and Kochanek (2009); Ivanov et al. (2016)).

Most important is the fact that the observation of the AGN radiation variability is the result of the accretion rate variability in the SMBH at the center of its host galaxy. Among the most well-known scenarios of the basic structure of the accretion disk, which can explain the AGN strong and rapid variability, are the models of magnetically elevated (or “thick”) disks (see Dexter et al. (2014); Begelman et al. (2015); Begelman and Silk (2017); Dexter and Begelman (2018)), the transitions of the accretion state (see Noda and Done (2018); Ruan et al. (2019); Graham et al. (2019)), instability arising as a result of the magnetic moment near the inner stable circular orbit around the accretion disk (see Stern et al. (2018); Ross et al. (2018)) and the misaligned disks (see Nixon et al. (2012, 2013); Nealon et al. (2015); Nixon and King (2016); Pounds et al. (2017, 2018); King and Nixon (2018)) and disk wind models (see Elitzur and Shlosman (2006); Elitzur and Ho (2009); Elitzur et al. (2014); MacLeod et al. (2019)).

Our disk wind model is very different from the known scenarios of the above-mentioned models.

We have shown that the existence of toroidal magnetic processes in the accretion disk is the result of the fundamental holographic principle of quantum gravity and, consequently, the existence of the generalized thermomagnetic EN effect in the BH “tachocline”. It is known that a toroidal (i.e. azimuthal) magnetic field is subject to both Parker (see Parker (1966, 1967, 1969a, 1979a)) and rotational shearing instabilities (differential rotation; see Velikhov (1959); Chandrasekhar (1960); Balbus and Hawley (1991)). The most important fact is that both amplification mechanisms operate consistently and at the same time (see e.g. Foglizzo and Tagger (1994, 1995)), depending on such differential rotation forces, which turn the accretion disk into a whirl, drawing matter into a BH, and rousing winds that blow some of it out.

This means (see Eq. (159)) that at the high density of DM the magnetic driving mechanism creates an accretion disk wind, which is equatorial (see Fig. 26c) and occurs in soft states of BHB bursts (see e.g. a similar model by Chakravorty et al. (2016)). Alternatively, at the low density of ADM, the magnetic pressure is rather high and, consequently, the magnetic driving mechanism creates a weak or less noticeable wind in hard states, which characterizes the appearance of jets from a BH.

Hence, we understand that the force of the magnetic driving mechanism in the disk, connected by hydrostatic equilibrium, is weak or strong when the vertical size of the horizontal magnetic field from the center of the disk is inversely proportional to the density (i.e. the sum of gravitation, thermal gas pressure, CR gas pressure and DM halo density (see e.g. Rodrigues et al. (2015))) of the gas in the disk (see e.g. Fig. 2 in Parker (1966), Fig. 3 in Parker (1967)). In other words, the larger the toroidal field, the smaller the vertical size of the horizontal magnetic field from the middle of the disk, and vice versa.

This raises the question of how the observational data on AGN (or jet) variability, which theoretically anticorrelates with a variation of the toroidal magnetic field in the accretion disk, will be related to the observational data on variations in the accretion rate or e.g. the magnetic disk winds? In our opinion, despite the elusiveness of direct observations, some ideas, for example, of Rodrigues et al. (2015), about the possible observation of Parker loops from synchrotron radiation of galaxies near their edges, or from the Faraday rotation, or the idea of Chakravorty et al. (2016) on the possible observable difference between winds and jets from accretion disks can provide the indirect observational support.

In this sense, we are very interested in obtaining the indirect observations of anticorrelation between winds and jets from accretion disks. The point is that the modulation of a ADM halo leads to the situation when the high density of ADM corresponds to the disk wind and high accretion rate, while the low density of ADM corresponds to the low accretion rate and, as a result, to the jets from the BH. As we understand, in order to get a generalized picture of the connections between winds and jets from accretion disks, it is necessary to obtain the indirect observation, which physically explains the remarkable connection between all mentioned systems: the modulation of the ADM halo at the center of the galaxy $\rightarrow$ waves of the vertical gravitational density from the disk to the solar neighborhood $\rightarrow$ variations (cycles) of sunspots $\rightarrow$ the variability of the local positions of orbital S-stars near the BH.

Our plan is the following. The main goal is to show that the observed variability of the local positions of the S-stars orbiting near the BH is an indicator (or dynamic probe) of the disk wind speeds variation or, equivalently, of the accretion rate variations of the BH. The intermediate goal is to briefly discuss the formation of the elliptically orbital distributions and periodic cycles of the S-stars, which are located in 0.13 ly $\approx$ 0.04 pc (see Fig. 26c) from the BH.

Here we present a detailed analysis of the effects of DM capture and energy transport on the structure and evolution of main-sequence B-stars, specifically those which might exist at the GC. First we need to highlight some important modulation properties of the S-stars and ADM around the BH:

$\bullet$ The greatest capture happens when the star is farthest from the center of the galaxy, at apoapsis. This is because it slows down relative to the ADM halo and achieves a significant capture rate for a time before turning back towards the BH. By the time it reaches periapsis, the star is moving so quickly that the capture is essentially zero, regardless of how high the ADM density is.

$\bullet$ When the S-stars, and especially, for example, S102 and S2, approach (retreat from) the center of the Milky-Way, at periapsis (apoapsis), the ADM density increases (decreases) and, thus, increases (decreases) the baryonic sector in the subparsec region near the SMBH at the GC. As a result, the variability of the ADM density and the baryonic matter density are determined by the variability of the gravitational potential around the BH, which is controlled by tidal interactions with other galaxies in the form of the Virgo-like cluster (see e.g. Semczuk et al. (2016)).

We have found that, in contrast to the notable paper by Merritt et al. (2009), the presence of the intermediate mass BH (IMBH; see evidence by Takekawa et al. (2019)) orbiting inside a nuclear star cluster at the GC can effectively help (by means of the AR-CHAIN ​​code (Mikkola and Merritt, 2008) and the “indirectly observable” ADM halo density modulation in the center of the Milky Way) to randomize the orbits of S-stars near the SMBH, converting the initially thin co-rotating disk (see the warped or mini-disk in Fig. 26c; see also Chen and Amaro-Seoane (2014, 2015)) into the almost isotropic distribution of stars moving on eccentric orbits around the SMBH (see Fig. 32). Here the word “almost”, as we believe, practically overturns the essence of understanding of the physics of randomizing the orbits of S-stars near the SMBH, since the initial distribution of stars is somewhat ad hoc, and the evolution to the distribution of thermal eccentricity itself occurs at the even more special time scale.

First of all, the randomization of the orbital planes requires $\leqslant 20~{}Myr$ (see Šubr, Fragione and Dabringhausen (2019)) if the IMBH mass is $(3.2\pm 0.6)\times 10^{4}~{}M_{Sun}$ (see Takekawa et al. (2019)) and if the orbital eccentricity is $\sim 0.7$ or greater. So, this means that in our view of the S-stars randomization, the final distribution of the main semiaxes of star orbits does not depend on the estimated size of the IMBH orbit (see Merritt et al. (2009)) or, for example, on the special orientation of the binary orbit (see Rasskazov and Merritt (2017)), but depends on the “indirectly observable” density of the DM halo at the center of the Milky Way. Its time scale modulation is determined by the time scale of the S-stars periods through the variability of the gravitational potential at the GC, which is controlled by means of the “observed” disk wind rates variations or, equivalently, the accretion rate variations of the BH.

Let us now return to the question posed above. If we take into account the evolution of the SMBH-IMBH binaries through the case of the ejection of high-velocity S-stars (Fig. 32a) and B-type hypervelocity stars (Fig. 32b), i.e. the so-called IMBH slingshot, then how is the time scale of the S-stars periods (through the variability of the gravitational potential at the center of the galaxy) driven and controlled by the “observable” variations of the disk wind speeds or, equivalently, the variations of the BH accretion rate?

It works as follows. Among the chaotically oriented orbits of S-stars, some of them which are oriented near the fundamental plane of the GC have a direction along the accretion to the SMBH. When an elliptically orbital star moves from the BH periapsis (at the high speed the capture of ADM is almost zero) to apoapsis (when the speed is lower, the capture of ADM is not too small), it means that the point of apoapsis is determined by the condition of hydrostatic balance, at which the deceleration of the S-star must be identical (in the absolute value) to the deceleration of the disk wind, or equivalently, the lower BH accretion rate. Conversely, the increase in the speed of an elliptical-orbit star from apoapsis to periapsis assumes the higher speed of the disk wind, or equivalently, the higher rate of the BH accretion. As a result, the periods of variability in the disk wind speed or accretion rate are an indicator of the ADM variability, and consequently, an indicator of the periods of S-stars, e.g. S102 and S2 with periods of about 11 and 16 years (see Fig. 25a,b).

It also means that among the isotropically distributed S-stars there are such stars that, although they have random orientations, do not have random velocities and periods moving along eccentric orbits near the fundamental plane of the GC (see Fig. 32a).

A unique result of our model is the fact that the periods, velocities and modulations of S-stars are a fundamental indicator of the modulation of the ADM halo density at the center of the Galaxy, which closely correlates with the density modulation of the baryon matter near the SMBH. If the modulations of the ADM halo at the GC lead to modulations of the ADM density on the surface of the Sun (through vertical density waves from the disk to the solar neighborhood), then there is an “experimental” anticorrelation connection between the modulation of the ADM density in the solar interior and the number of sunspots. Therefore, this is also true for the relationship between the periods of S-star cycles and the sunspot cycles (see “experimental” anticorrelated data in Fig. 25a,b).

5 Summary and Outlook

In the given paper we present a self-consistent model of the axion mechanism of the Sun luminosity variations, in the framework of which we estimate the values of the axion mass ($m_{a}\sim 3.2\cdot 10^{-2}~{}eV$) and the axion coupling constant to photons ($g_{a\gamma}\sim 4.4\cdot 10^{-11}~{}GeV^{-1}$).

This result for the first time (see Rusov et al. (2015)) suggested that the existence of photons of axion origin in a virtually empty MFT, which is anchored to the tachocline, is the consequence of the Parker-Biermann cooling effect, which causes the strong magnetic pressure in the twisted magnetic tubes (see Fig. 5b and Fig. 2). This result also suggests that the existence of the magnetic field of the virtually empty anchored flux tube should be a consequence of the existence of the strong magnetic field in the tachocline, which, in contrast to the solar dynamo action, is much larger than $10^{5}~{}G$.

This means that the existence of the tachocline, on the one hand, must be predetermined by the existence of the holographic principle of quantum gravity, and on the other hand, it gives rise to the BL holographic mechanism (see Fig. 4b), which, unlike the solar dynamo models, generates a strong toroidal magnetic field by the thermomagnetic EN effect in the tachocline ($\sim 10^{7}~{}G$ (Eq. (27)); see also Fig. 3).

Leaving aside some conclusions from the interesting solved problems (the nature of the toroidal magnetic field of the tachocline in the interior of the Sun (see Sect. 3), the universal model of antidynamo flux tubes and the phenomenon of DM solar axions (see Sect. 10), the phenomenon of magnetic reconnection in the lower layers of the magnetic tube and observable tendencies of the tilt angle of Joy’s Law (see Sect. 11), the solar axion and coronal heating problem solution (see Sect. 3.2.2), the chronometer of DM hidden in the Sun core (see Sect. 23), the modulation of the ADM density as a “clock” regulating the tempo of the solar cycle and the mechanism of ADM variations around the BH (see Sect. 4)), we focus on the fundamental connection between the Meissner-like and BZ-like effects on the BH boundary, which resolves the information loss paradox, a long-standing problem in theoretical physics.

Understanding the physics of this topic reinforces a very surprising and intriguing question about whether the theory of the complete restoration of information loss is different or not in our model and the known Hawking model (Hawking, 2015a; Hawking et al., 2016). Our answer is yes and no!

At the conference in Stockholm on August 28, 2015, Stephen Hawking announced that he had solved the information paradox. According to Hawking (Hawking, 2015b, a; Hawking et al., 2016), the AdS/CFT correspondence showed no loss of information, where all information about matter and energy in the three-dimensional volume of the BH is encoded in the form of a hologram on its two-dimensional surface of the event horizon (see ’t Hooft (1993), Susskind (1995), Maldacena (1999), Hanada et al. (2014)). It resembles the analog of the famous photo-hologram by Gabor (1948), which is a photo of a certain type that generates a three-dimensional image when it is well-lit; all information describing a three-dimensional scene is encoded in a pattern of light and dark areas inscribed on a two-dimensional film. This means that the possible description and solution of the “encoded” holographic two-dimensional “film-plate” is predetermined by the fact that the problem of the information paradox, known as the apparent contradiction between the general theory of relativity and quantum mechanics, completely disappears. This approach ultimately leads to the holographic principle of quantum gravity, which can be generalized for any physical system occupying space-time. For example, in the holograms of the Universe, a BH, white dwarfs, neutron stars or, surprisingly, in the hologram of the Sun tachocline!

Hawking’s main result is the fact that the paradox of information loss can be resolved by supertranslations of the horizon of BHs forming a hologram of incoming particles (Hawking, 2015b, a), where the information itself can be later completely restored, albeit in the chaotic form because of the radiation emitted during the quantum evaporation of the BH (Hawking, 2015b, a; Hawking et al., 2016) predicted forty years ago by Hawking. This is due to the fact that the concept of the asymptotic isometry of the AdS space (see Brown and Henneaux (1986)), which consists only of the supertranslation of time for all Bondi-van der Burg-Metzner-Sachs models (Bondi et al. (1962), Sachs and Bondi (1962)), has long attracted considerable attention from the point of view of the AdS/CFT correspondence (Maldacena (1999)), in which supergravity in the AdS space is equivalent to the CFT at the boundary and gives a lot of information about the entropy problem in BHs and the strong connection between the gauge theory and the string theory (see Strominger and Vafa (1996), Witten (1998), Strominger (1998); Strominger and Zhiboedov (2016)).

But the most surprising is the fact that the information stored in supertranslation with the help of a shift in the horizon caused by ingoing particles is returned via outgoing particles in a scrambled chaotically useless form. In other words, we can say that the volume of recovered information is a complete “book” of information, but, unfortunately, it is not readable.

Oddly enough, this raises the question of whether a complete “book” of information can somehow be read. The answer is very simple: “yes”!

In this regard, we consider a rather unexpected connection between the theory of the EN effect, the second law of thermodynamics, and the paradigm of fuzzball complementarity on the horizon of BH events.

It is known that the AdS/CFT correspondence brought the holographic principle to the central stage of the string theory and, thus, can help to understand the nature of BHs. On the other hand, we know that several years ago, Mathur and his group discovered (see e.g. Mathur and Turton (2012, 2014a, 2014b, 2018) and Refs. therein; see also Guo et al. (2018); Mathur (2005, 2009a, 2009c, 2010, 2011, 2012a, 2012b, 2012c, 2015, 2017a)) that in the string theory BHs have a completely different structure: instead of vacuum and the singularity of a BH, in which the region just outside the horizon is the Rindler space (see Fig. 3a in Mathur (2017a)), there are microstates of the string theory that are known as fuzzballs of mass $M$ (with surface at $r=2GM+\varepsilon\equiv r_{b}$); they do not have a horizon or singularity, and do not collapse the spheres into a BH. It should be remembered that because of the altered vacuum fluctuations, the region near the fuzzball boundary is called the pseudo-Rindler space (see Fig. 3b in Mathur (2017a)).

The most interesting is the fact that this fuzzball, according to Mathur (2009c); Mathur and Turton (2014a, b); Mathur (2017a, 2018b, 2018a) is radiated at the same temperature as the Hawking BH. But the radiation emerges from the fuzzball surface in the same way as for any other normal body (like burning a lump of coal (see Mathur (2009c, b); Alonso-Serrano and Visser (2016)), and therefore the radiation (and its thermal spectrum) carries the information present on this surface. This eliminates the information paradox (see e.g. Mathur (2009b, 2017a, 2017b); Alonso-Serrano and Visser (2018)).

Strengthening the paradigm of fuzzball, i.e. the microstates of BHs (see e.g. Fig. 33a), we note the fact that unlike the well-known Hawking’s problem of quantum corrections (see e.g. Hawking’s argument as ’theorem’ (Mathur, 2009c, 2017b)) and the firewall argument with the help of the so-called ‘validity of effective field theory’ (Almheiri, Marolf, Polchinski, Stanford and Sully (2013)), an infalling shell-object never falls into the trap of its own horizon, and thus avoids any problem with causality.

Hence, it follows that, according to Mathur and Turton (2014a, b), the information paradox (and causality) is solved with the help of the construction of fuzzball in the string theory, which, without resorting to new physics, gives a real degree of freedom on the fuzzball surface and not a virtual degree of freedom on the horizon. These degrees of freedom should play two roles: 1) radiate Hawking quanta at $E\sim T$ (see Fig. 33b) and 2) account for any complementarity through collective oscillations generated by hard impacts of $E\gg T$ quanta (see Fig. 33c), where $E$ is the energy related to the physical process and $T$ is the Hawking temperature, both measured at infinity.

The key point is that in the string theory the fuzzball surface differs from the microstate to the microstate (see Fig. 33a) or from the wave function of the ball to the wave function (see Fig. 33a) that propagates through this space without a horizon or singularity and thus gives birth to two processes: (i) the Hawking radiation ($E\sim T$), which is very sensitive to microstates and carries the microstate information, and (ii) only complementarily excited collective modes ($E\gg T$), which mimic free infall and, as a consequence, are relatively insensitive to the choice of microstates and can have any possible complementary description with an information-free horizon, where, according to Mathur (2015), the spectrum of excitations of the fuzzball surface agrees, to a good approximation, with the spectrum of infalling modes in the traditional BH. This solves the information paradox.

It also means that Hawking radiation ($E\sim T$), which is very sensitive to microstates, will carry information about the microstate through the thermomagnetic EN effect in the tachocline of the black hole. From this we understand that it is necessary to replace the fuzzball surface with the EN-fuzzball surface, which is now clear in Fig. 29a.

Unlike the information about the BH horizon by Hawking (2015a); Hawking et al. (2016), the recovered information of the BH fuzzball found by Mathur and his group (see e.g. Mathur and Turton (2014a, b); Mathur (2011, 2015, 2017a, 2017b); Guo et al. (2018) is surprisingly a completely readable “book”. However, although at this stage these two groups cannot prove that the recovered information is true, we need to understand how one of these information models of a BH can be associated with some estimates of the already known measured signals of deviations from a traditional BH.

In this regard, we are interested in the observable astrophysical signs of quantum gravity near BHs (see e.g. Cardoso and Pani (2017)) in which some features of the data already obtained can be interpreted as evidence of a gravitational echo (see Cardoso et al. (2016); Abedi et al. (2017)). This means that in our case, in addition to measurable signals exhibiting significant deviations from classical BH evolution in the near-horizon region, it can give unique observable signs of the formation of fuzzball, where the collapsing shell suffer “entropic enhanced tunneling” into fuzzballs (Mathur, 2017a), which usually breaks symmetry and is expected to result in bursts of gravitational and electromagnetic radiation (see Hertog and Hartle (2017)).

In our view, these quantum reflected echo signals (see Fig. 1 in Abedi et al. (2017)) are possible not only from the BH mergers, but also from the single SMBHs, where the DM (and, accordingly, baryons) modulation determines the formation and existence of the galactic gravitational waves (see Eq. (2.1) in Allen and Romano (1999); Eq. (2) in Abbott et al. (2016b); Eqs. (4.1) and (4.3) in Raidal et al. (2017); Romano and Cornish (2017); Kovetz (2017)).

It is very important to understand that the modulations of DM (and, accordingly, baryons) predetermine not only the emergence of galactic gravitational waves, but also the modulation of gravitational waves, in which gravitational theories predict the possibility of creating matter associated with the geometry-matter connection in cosmology and astrophysics (see Parker (1968, 1969b); Zel’dovich (1970); Zel’Dovich and Starobinskiǐ (1972); Fulling et al. (1974); Parker and Toms (2009); Parker (2009)).

It means that the particle creation corresponds to the irreversible energy flow from the gravitational field to the created matter constituents, with the second law of thermodynamics requiring that space-time transforms into matter (see Harko (2014)). We are interested in the irreversible thermodynamic process that is a direct consequence of the so-called non-minimal coupling of curvature-matter, induced in this case by quantum fluctuations of the gravitational metric (see Harko (2014); Harko and Lobo (2013); Harko et al. (2015); Liu et al. (2016)), as initiated by Yang (2016); Dzhunushaliev et al. (2015); Dzhunushaliev (2015).

The essence of the remarkable idea of ​​Dzhunushaliev and his group is as follows. They show that if the metric in quantum gravity can be represented as a sum of the classical and quantum parts (see Eq. (9) in Dzhunushaliev et al. (2014)), then such a gravitating physical system looks like modified gravity (see Eq. (21) in Dzhunushaliev et al. (2014)). This is, on the one hand, due to the nonzero mathematical expectation of the quantum part of the metric, and on the other hand, there is a non-minimal interaction between matter and gravity. The general approach of quantum corrections arises both from the nonperturbative quantization of the metric (Dzhunushaliev and Folomeev, 2014) and from the perturbative quantization of various types of fields (scalar, electromagnetic, QCD) (Dzhunushaliev and Folomeev, 2015).

Based on the idea of Dzhunushaliev and his group, Harko and his group made a very important assumption (see Harko (2014); Harko and Lobo (2013); Harko et al. (2015); Liu et al. (2016)) that the coupling coefficient between the metric and the average value of the quantum fluctuation tensor is a scalar field with a non-vanishing self-interaction potential and a simple scalar function. One of the most important results of this assumption is the fact (see Liu et al. (2016)) that it is the self-interaction potential scalar field that was assumed to be of Higgs type (Aad et al., 2015).

Hence, there is an intriguing problem of the Higgs-like boson. In our work (see the topic “Multiparticle fields and the Higgs mechanism” in Sect. 4), we have shown that, given the experimental data of Patrignani and Particle Data Group (2016), it is possible to evaluate the existence of a scalar Higgs field in internal indices that do not interact for electroweak and strong interactions. At the same time, such a scalar field can contribute to the gravitational field through the energy-momentum tensor. Thus, the Higgs bosons, scalar in internal indices, can be considered a candidate for a part of DM!

As a result, we understand that the Higgs-like particles do not have any additional fundamental interaction and contain the part of VM – the Higgs-like baryon particles, and the part of DM – Higgs-like DM (see Fig. 28b). That is why the effects of Higgs-like particles can, on the one hand, be identical to the effects of EN-fuzzball (see Fig. 29a), and, on the other hand, when they decay into two photons, they will be identical to the effects of BZ-fuzzball, which is now also clear from Fig. 29b.

Here we are interested in the problem of the Higgs-like particles decay channel into two photons (see ATLAS Collaboration (Aad et al., 2015) and Patrignani and Particle Data Group (2016)) and the link (see the topic “•••••” in Sect. 4) between the physics of radiation of gravitational waves and the generalized thermomagnetic EN effect of DM around a BH. Since we know (see the topics “•••” and “••••” in Sect. 4) that the EN effect near the “fuzzball” surface of a BH is absolutely identical to the Higgs phenomenon in particle physics (see Wilczek (2000, 2005)), i.e. quantum phenomenon of particles of DM, there appears a wonderful way to solve the problem of the Higgs-like particles decay channel near the “BZ fuzzball” surface of a BH.

In the beginning, we demonstrate the simple physics of this problem. (1) If the Higgs-like particles are near the surface of the “fuzzball” of a BH, then we call it EN fuzzball, at which unitary Hawking radiation is emitted ($E\sim T$; see Fig. 33b). (2) If the decay channel of Higgs-like particles into two photons appears near the surface of a BH fuzzball, then we call it EN fuzzball, in which photons with energy $E\gg T$ excite the collective dynamics of the fuzzball. This is the solution of the problem of BZ fuzzball complementarity (see Fig. 33c).

Here the question arises as to how the theoretical solution of the fuzzball complementarity problem (see Fig. 33) is related to the known experimental data, for example, using signals from gravitational waves as signatures of quantum gravity towards the horizon of BHs. In this sense, we are interested in the well-known observation of the space-time image of the echo signals of gravitational waves on the EN fuzzball surface (or the identical Meissner-like fuzzball (see Fig. 27a)) on the stretched horizon after the BH accretion event (see Fig. 29a; adopted from Abedi et al. (2017)).

Based on the model of the fuzzball complementarity (see Fig. 29), we see that the echo waves, i.e. the gravitational waves trapped between the structure near the horizon and the angular momentum barrier (see Fig. 1 in Abedi et al. (2017)), are reflected on the EN fuzzball surface (see Fig. 29a), which is the result of the generalized thermomagnetic EN effect on BHs (see Fig. 27a) or, equivalently, the phenomenon of Higgs-like particles. On the other hand, if we assume that the gravitational waves penetrate the BZ fuzzball surface (see Fig. 29b) as a consequence of the BZ effect (see Fig. 27b), this means that due to the decay of the Higgs particles, photons with energy $E\gg T$, exciting the collective dynamics of the BZ fuzzball surface, effectively penetrate into it, and thus the gravitational waves also pass there.

Here comes the second question about what the physics of the BZ-like effect is. Until now, it is believed that the central BH engine, which is the source of GRBs, includes the electromagnetic-magnetic extraction of the BH rotational energy, proposed by Blandford and Znajek (1977). Unlike the well-known discussions in the GRB community (see e.g. Lee et al. (2000); Wang et al. (2002); McKinney (2005); Komissarov and Barkov (2009); Nagataki (2009); Contopoulos et al. (2017)), we use the very interesting interpretation of Salafia (2017) about the properties of a particular short GRB (SGRB), whose progenitor is the merger of two neutron stars in the form of a BH, or the BH and neutron star, and the remarkable first observation of gravitational waves from the fusion of a binary neutron star with a SGRB-GW connection (Abbott et al., 2017c, a, b). The two phenomena are thus inseparable (see e.g. Salafia (2017)).

Skipping the complex but very beautiful task of the first observations of a kilonova (which follows the initial fuzzball), the UV/Optical/Infrared emission from the expanding material ejected during the merger stages and coalescence driven by the nuclear decay of unstable nuclei synthesized by the $r$-process (see Salafia (2017) and Refs. therein) and/or the creation of new (dark) matter related to the geometry-substance coupling induced by the quantum fluctuations of the gravitational metric (see Liu et al. (2016) and Refs. therein), we note that the physics of two undivided observations of SGRB and GW is the theoretical basis of the fuzzball complement model (see Fig. 29), which as a consequence of holographic quantum gravity is predetermined by the BZ-like effect (see Fig. 27b) on BHs.

We will discuss the implications of our findings in a forthcoming publication.

And finally, let us emphasize one essential and most painful point of this paper. This is the key problem of the holographic principle of quantum gravity, on the basis of which the thermomagnetic EN effect predetermines the possibility of observational measurements of magnetic fields between the two-dimensional surface of the tachocline and the three-dimensional volume of cores in compact objects – our Sun, magnetic white dwarfs, accreting neutron stars and BHs. For example, with the help of the thermomagnetic EN effect, a simple estimate of the magnetic pressure of an ideal gas in the tachocline of e.g. the Sun can indirectly prove that by using the holographic principle of quantum gravity, the toroidal magnetic field of the tachocline accurately “neutralizes” the magnetic field in the Sun core. This means that the holographic BL mechanism is the main process of regenerating the primary toroidal field in the tachocline and, as a consequence, the formation of buoyant toroidal tubes of magnetic flux at the base of the convective zone, which then rise to the surface of the Sun. Hence, two phenomena – DM in the form of ADM and axions – ground the understanding of the nature of the holographic principle, starting, for example, from the solution of the problem of antidynamo, sunspots and coronal heating and, oddly enough, to the complete restoration of information loss of the BH, which, although it differs from the well-known Hawking model, does not contradict the experimental observations.

Here the question arises: is it true, is the reason simple or not? Or is it just the guessed rules of calculation that do not reflect any real nature of things, or “…when we would backward see from what region of remoteness the whatness of our whoness hath fetched his whenceness…” (James Joyce. Ulysses, Episode 14)?

Appendix A X-ray coronal luminosity variations

The X-ray luminosity of the solar corona during the active phase of the solar cycle is defined by the following expression:

[TABLE]

In the quiet phase it can be written as

[TABLE]

Then, integrating the blue curve in Fig. A.1 for $\left(L_{corona}^{X}\right)_{max}$ and the cyan curve for $\left(L_{corona}^{X}\right)_{min}$, we obtain

[TABLE]

So it may be derived from here that the Sun luminosity is quite low in X-rays (A.3), typically (see analogue in Rieutord (2014))

[TABLE]

but it varies with the cycle (see blue and cyan lines in Fig. A.1a) as nicely shown by the pictures obtained with the Yohkoh satellite (see analogous Fig. 4 in Rieutord (2014)).

And finally, it may be supposed that X-rays, propagating from the tachocline towards the photosphere, interact with the charged particles via the Compton scattering, but only outside the magnetic tubes. The axion-originated X-ray radiation channeling inside the magnetic tubes does not experience the Compton scattering up to the photosphere (Figure A.1).

Appendix B Explanation of the high X-ray intensity bands widening near the Yohkoh image edges

It is interesting to note that the bands of high X-ray intensity on Yohkoh images deviate from the solar parallels (Fig. 13b in Zioutas et al. (2009)). This is especially the case near the edges of the visible solar disk.

This effect can be explained graphically by means of Figs. B.1 and B.2. These figures show the schematic concept of the Sun image formation on the Yohkoh matrix. Fig. B.1 shows the Sun from its pole.

Let us choose three sectors of equal size on the surface of the Sun (Figs. B.1, B.2). The areas of the photosphere cut by these sectors are also equal ($S_{0}=S_{1}=S_{2}$). However, as it is easily seen in the suggested scheme, the projections of these sectors on the Yohkoh matrix are not of equal area (Figs. B.1, B.2). The $S^{\prime}_{1}$ and $S^{\prime}_{2}$ projections areas are much less than that of the $S^{\prime}_{0}$ projection (Fig. B.1). This means that the radiation emitted by the sectors $S_{1}$ and $S_{2}$ of the solar photosphere and captured by the satellite camera will be concentrated within lesser areas (near the edges of the solar disk) than the radiation coming from the $S_{0}$ sector (in the center of the solar disk). As a result, the satellite shows higher intensity near the image edges than that in the center, in spite of the obvious fact that the real radiation intensity is equal along the parallel of the Sun.

Therefore, because of the system geometry, the satellite tends to “amplify” the intensity near the image edges, and the areas that correspond to the yellow and green areas in the center (Fig. 13b in Zioutas et al. (2009)) become red near the edges, thus leading to the visible widening of the high intensity bands.

The particularly high radiation intensity near the very edges of the visible solar disk, observed even during the quiet phase of the Sun Fig. 13a in Zioutas et al. (2009), indicates a rather “wide” directional radiation pattern of the solar X-rays.

Let us make a simple computational experiment. We choose a sphere of unit radius and spread the points over its surface in such a way that their density changes smoothly according to some dependence on the polar angle ($\theta$). The azimuth angle does not influence the density of these points. For this purpose any function that provides a smooth change of the density will do. For example, this one:

[TABLE]

Here we take $\rho_{0}=3.5$ and $\rho_{max}=3$ in arbitrary units. The graphical representation of this dependence is shown in Fig. B.3. It yields the minimum density of points near the poles and the equator, and the maximum density of points near $\theta=60^{\circ}$ and $\theta=120^{\circ}$.

The polar angle $\theta$ was set in the range $[0,180^{\circ}]$ with the step of $1^{\circ}$. The azimuth angle $\varphi$ was set in the range $[0,180^{\circ}]$ (one hemisphere) with a variable step $\Delta\varphi$ representing the variable density, since the points density is inversely proportional to the step between them ($\Delta\varphi\sim 1/\rho$). We assume that

[TABLE]

The values of $\Delta\varphi_{0}=3.5^{\circ}$ and $\Delta\varphi_{max}=3^{\circ}$ were chosen arbitrarily. So,

[TABLE]

From Eq. (B.3) it is clear that the minimum step was 0.5*∘* and the maximum step was 6.5*∘*. Apparently, the more is the step, the less is the density (near the poles and the equator) and vice versa, the less is the step, the more is the points density. This was the way of providing the smooth change of the points density by latitude (along the solar meridians).

Obviously, this forms the “belts” of the high density of points along the parallels. The projection of such a sphere on any plane perpendicular to its equator plane will have the form shown in Fig. B.4a. As it is seen in this figure, although the density does not depend on the azimuth angle, there are high density bands widening near the edges of the projected image. These bands are similar to those observed on the images of the Sun in Fig. B.4b.

Let us emphasize that the exact form of the dependence (B.2) as well as the exact values of its parameters were chosen absolutely arbitrarily for the sole purpose of the qualitative effect demonstration. They have no relation to the actual latitudinal X-ray intensity distribution over the surface of the Sun.